GRAFT-ATHENA: Self-Improving Agentic Teams for Autonomous Discovery and Evolutionary Numerical Algorithms

# GRAFT-ATHENA: Self-Improving Agentic Teams for Autonomous Discovery and Evolutionary Numerical Algorithms
Source: [https://arxiv.org/html/2605.11117](https://arxiv.org/html/2605.11117)
###### Abstract

Scientific discovery can be modeled as a sequence of probabilistic decisions that map physical problems to numerical solutions\. Recent agentic AI systems automate individual scientific tasks by orchestrating LLM\-driven planners, solvers, and evaluators\. Each method is a combination of methodological actions, with many viable combinations for any given problem and structural dependencies between choices\. However, existing frameworks treat each problem in isolation, with no shared substrate to accumulate methodological experience across domains\. Here we show that GRAFT\-ATHENA, a self\-improving agentic framework, learns from past problems and autonomously expands its own action space across diverse domains\. GRAFT \(Graph Reduction to Adaptive Factored Trees\) projects combinatorial decision spaces into factored probabilistic trees in which each method is a single path, taking the parameter footprint from exponential to linear\. In the lineage of classical Bayesian networks, the factorization is anII\-map of the policy, and the resulting paths embed as unique fingerprints in a metric space whose closeness lets each new problem learn from similar past ones\. On canonical physics\-informed machine learning \(PIML\) benchmarks, GRAFT\-ATHENA reaches near\-machine\-precision accuracy, surpassing human and prior agentic baselines, and on production solvers, it tackles complex engineering problems such as reconstructing Mach\-10 flow over the Apollo Command Module from a 1968 report and recovering shear\-thinning blood\-cell rheology\. Notably, the system grows its own knowledge substrate, autonomously proposing regularization constraints for ill\-posed inverse problems and discovering new numerical methods such as a spectral PINN with exponential convergence\. These results provide a foundation for autonomous laboratories that grow more capable with every problem they solve\.

###### keywords:

Agentic AI , Scientific Computing , Scientific Machine Learning , Knowledge Graphs , Factored Decision Trees , Continual Learning , Autonomous Scientific Discovery

††journal:arXiv\\affiliation

\[inst1\]organization=Division of Applied Mathematics, Brown University,city=Providence, postcode=02912, state=RI, country=USA

label2label2footnotetext:Corresponding author: george\_karniadakis@brown\.edu## 1Introduction

Autonomous scientific computing is moving beyond code generation and fixed\-solver execution toward agents that choose methods, run them and evaluate outcomes\. Recent systems carry such tasks end to end, from problem specification through experiment proposal to refinement, on benchmarks ranging from materials discovery to drug repurposingGottweiset al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib792)\], Ghafarollahi and Buehler \[[2025](https://arxiv.org/html/2605.11117#bib.bib19)\], Villaescusa\-Navarroet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib795)\], Buehler \[[2025](https://arxiv.org/html/2605.11117#bib.bib805)\], Stewartet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib800)\], Stewart and Buehler \[[2026](https://arxiv.org/html/2605.11117#bib.bib797)\], Ni and Buehler \[[2026](https://arxiv.org/html/2605.11117#bib.bib806)\], Jiang and Karniadakis \[[2026](https://arxiv.org/html/2605.11117#bib.bib27)\]\. Across these efforts, the goal is an agent that grows more capable with every problem it solves\. A laboratory that improves every time it runs is a categorically different computational tool, not merely a faster one\. Realizing this aspiration requires more than stronger planners: it requires an organizing substrate that records method choices, outcomes, and reusable experience across problems\. The missing substrate is one that records, organizes, and reuses the methodological experience of past problems\. Most current systems, including ATHENA, are organized around successful execution or refinement within a given problem rather than around a shared, measurable action space that persists across problem families\. This is a natural stage in the field: planners, coding agents, scientific\-agent frameworks, and large solution sweeps have shown substantial task\-level capability, but they usually store experience as text, trajectories, or artifacts rather than as reusable method neighborhoods\. The shared limitation is structural rather than incidental: the action space inside an LLM\-driven agent is usually implicit, namely whatever the model can emit inside a prompt, and is therefore not enumerable, navigable, or measurable\. Without such a space, distances between strategies cannot be defined, reward\-weighted priors cannot be assigned, and coverage cannot be verified; each new problem consequently begins cold, and success on problemnndoes not propagate to problemn\+1n\+1Toscanoet al\.\[[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\], Deotaleet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib807)\], Fenget al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib798)\], Ghafarollahi and Buehler \[[2025](https://arxiv.org/html/2605.11117#bib.bib19)\], Jeonet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib799)\], Wanget al\.\[[2026a](https://arxiv.org/html/2605.11117#bib.bib801)\], Subramaniamet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib791)\]\.

This missing substrate limits not only what an agent can remember, but also the kind of reasoning it can support\. The relevant distinction is between association, which links similar cases; intervention, which tests chosen actions; and counterfactual reasoning, which asks how the outcome would have changed under an unchosen action\. Pearl’s ladder of causation formalizes this hierarchy and gives a useful diagnostic for agentic scientific computingPearl \[[2018](https://arxiv.org/html/2605.11117#bib.bib17),[2019](https://arxiv.org/html/2605.11117#bib.bib9)\]\. Vanilla LLM and retrieval pipelines mainly operate by association, pattern\-matching on what looks likeXXwithout an action\-outcome handle that supports transfer beyond surface similarityWuet al\.\[[2023a](https://arxiv.org/html/2605.11117#bib.bib509)\]\. Most current agentic frameworks for scientific computing add intervention through proposer\-critic loops that act, observe, and correct within a single problemGhafarollahi and Buehler \[[2025](https://arxiv.org/html/2605.11117#bib.bib19)\], Jeonet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib799)\], Wanget al\.\[[2026a](https://arxiv.org/html/2605.11117#bib.bib801)\], Subramaniamet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib791)\], Deotaleet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib807)\], Georgievet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib25)\], Xuet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib796)\], Yanget al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib803)\],[Luoet al\.](https://arxiv.org/html/2605.11117#bib.bib794)\. Counterfactual reasoning remains out of reach today because current systems rarely expose a persistent, inspectable action space on which distances, priors, and alternative actions can be defined\.

ATHENAToscanoet al\.\[[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\]illustrates both the promise and the boundary of this intervention\-level regime\. Modeled as a contextual bandit, this framework used hierarchical teams of LLM agents to select actions, execute code, observe outcomes, and refine solutions within a problem, reaching state\-of\-the\-art performance on PIML benchmarks\. It also showed that expert methodological knowledge can help agent teams uncover exact solutions through symmetries or transformations and propose methods for cases such as inviscid Burgers with PINNs\. Its broader lesson was that expert knowledge is essential for agentic scientific computing\. In ATHENA, however, that knowledge lived mainly in system prompts, scaffolds, and run histories rather than in a measurable action space with a natural expansion rule\. The open prompt space allowed discovery, but new attempts and strong outcomes were not converted into reusable state\. The system therefore demonstrated the intervention loop, while still lacking a mechanism by which experience could accumulate and transfer systematically from one problem to the next\.

To address these limitations, we introduce GRAFT\-ATHENA, a self\-improving agentic framework built on a substrate that is explicit, navigable, growable, and shared across problems\. The substrate is produced by GRAFT \(Graph Reduction to Adaptive Factored Trees\), a construction that turns a directed acyclic graph of solver attributes and cross\-rules into a factored decision tree with a deterministic embedding into a metric space\. Each method is a single path through the action tree𝒯A\\mathcal\{T\}\_\{A\}, each problem a single path through its companion problem tree𝒯P\\mathcal\{T\}\_\{P\}, and the persistent memory𝒟\\mathcal\{D\}records every solved instance together with its observables and reward\. The locus of accumulated scientific experience is therefore the substrate, not the language model, since neighbors retrieved from𝒟\\mathcal\{D\}feed a reward\-calibrated prior over candidate methods on every new problem\. The framework supports association through memory, intervention through execution and repair, and the recorded action space needed for future counterfactual queries\.

GRAFT supplies the formal substrate\. It projects a DAG of solver attributes into a factored decision tree, retaining cross\-rules as local preconditions rather than folding every dependency into a single joint categorical\. This changes the policy footprint from exponential,∏j\|𝒜j\|\\prod\_\{j\}\|\\mathcal\{A\}\_\{j\}\|, to linear in the decision chains and rules,∑j\|𝒜j\|\+\|ℛ\|\\sum\_\{j\}\|\\mathcal\{A\}\_\{j\}\|\+\|\\mathcal\{R\}\|, while certifying the resulting factorization as anII\-map of the policyVerma and Pearl \[[1988](https://arxiv.org/html/2605.11117#bib.bib782)\]\. The same tree gives every problem and method a unique fingerprint, so distances between strategies are computable, and the nearest successful neighbors in𝒟\\mathcal\{D\}become a reward\-weighted prior for the next run\. The agentic system runs on this substrate in the same order a scientific problem enters the workflow\. Expansion and Construction teams first grow the action graph by ingesting solver documentation and validating each new branch with GRAFT\. A Formalization team then turns a free\-form user request into a well\-posed problem representation, auditing equations, boundary data, reductions, and identifiability before encoding it as a fingerprint on𝒯P\\mathcal\{T\}\_\{P\}\. Conditioned on that fingerprint, the Strategy team samples a method from𝒯A\\mathcal\{T\}\_\{A\}, the Implementation team realizes it as runnable code, and the Advisor scores the outcome, revises failed choices, or extends the action tree when the current vocabulary is insufficient\. Every attempt is committed to𝒟\\mathcal\{D\}, so within\-problem correction and cross\-problem accumulation are parts of the same loop\. The same machinery handles any solver whose user\-facing knobs admit a DAG description, exercised here on PDE solvers \(Nektar\+\+Cantwellet al\.\[[2015](https://arxiv.org/html/2605.11117#bib.bib14)\], Trixi\.jlRanochaet al\.\[[2021](https://arxiv.org/html/2605.11117#bib.bib220)\], PIMLToscanoet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib23)\]\) and on a non\-PDE solver \(LAMMPSThompsonet al\.\[[2022](https://arxiv.org/html/2605.11117#bib.bib63)\]for dissipative particle dynamics\)\.

The same machinery produces three categories of results\. On four canonical PIML benchmarks, GRAFT\-ATHENA reaches lower error floors than human and recent agentic baselines, including ATHENAToscanoet al\.\[[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\]\(Table[1](https://arxiv.org/html/2605.11117#S2.T1)\)\. The viscous Burgers case exposes the mechanism directly, because the prior retrieves a Reynolds\-number continuation from neighboring high\-Reynolds\-number runs and uses it to improve on the earlier agentic trace even at moderateRe\\mathrm\{Re\}\(§[2\.4](https://arxiv.org/html/2605.11117#S2.SS4)\)\. On real engineering targets, the system reconstructs Mach\-10 hypersonic flow over the Apollo Command Module from a 1968 NASA reportGriffith and Boylan \[[1968](https://arxiv.org/html/2605.11117#bib.bib16)\]\(§[2\.5](https://arxiv.org/html/2605.11117#S2.SS5)\) and recovers shear\-thinning red\-blood\-cell rheology in dissipative particle dynamics \(§[2\.6](https://arxiv.org/html/2605.11117#S2.SS6)\)\. On open\-ended discovery, it proposes regularizers for ill\-posed inverse problems \(§[2\.7](https://arxiv.org/html/2605.11117#S2.SS7)\) and designs a spectral PINN with exponential convergence \(§[2\.8](https://arxiv.org/html/2605.11117#S2.SS8)\), extending the action space and persisting the extensions in𝒟\\mathcal\{D\}\. Knowledge therefore accumulates monotonically, since every iteration, success or failure, is committed to the long\-term memory \(§[4\.1\.8](https://arxiv.org/html/2605.11117#S4.SS1.SSS8)\)\.

The contributions of this work can be summarized as follows\.

1. 1\.GRAFT turns a combinatorial scientific\-method space into a factored probabilistic decision tree with explicit dependency rules, an injective fingerprint embedding, a Jaccard metric, and a reward\-calibrated prior, with formal guarantees forII\-map certification, injectivity, and metric structure stated in Proposition[4\.1](https://arxiv.org/html/2605.11117#S4.Thmtheorem1), Proposition[4\.2](https://arxiv.org/html/2605.11117#S4.Thmtheorem2), and Proposition[A\.3](https://arxiv.org/html/2605.11117#A1.Thmtheorem3)\. ![Refer to caption](https://arxiv.org/html/2605.11117v1/Figures/Framework.png)Figure 1:GRAFT\-ATHENA system overview\.\(A\) Knowledge\-graph extension: an Expansion team distils a draft DAG and hints from solver documentation, methods papers, or a domain expert; a Construction team integrates the draft, classifying each hint as a cross\-attribute dependency or a general node decoration, with GRAFT \(Graph Reduction to Adaptive Factored Trees, §[2\.2](https://arxiv.org/html/2605.11117#S2.SS2)\) admitting only fragments that yield a rule\-preserving tree projection\. \(B\) Long\-term substrate: two knowledge graphs𝒢P\\mathcal\{G\}\_\{P\}\(problem space\) and𝒢A\\mathcal\{G\}\_\{A\}\(action vocabulary\), plus the memory𝒟=\{\(pj,mi,Oi,ri\)\}\\mathcal\{D\}=\\\{\(p\_\{j\},m\_\{i\},O\_\{i\},r\_\{i\}\)\\\}recording every solved instance with observablesOiO\_\{i\}and rewardrir\_\{i\}; new nodes leave past entries valid, so𝒟\\mathcal\{D\}grows non\-destructively\. \(C\) Per\-problem setup: the Formalization team turns a free\-form user query into a structured user request, guided by an analytical tree𝒯F\\mathcal\{T\}\_\{F\}\(a GRAFT projection of an analytical DAG𝒢F\\mathcal\{G\}\_\{F\}of simplification moves\) that scaffolds without constraining; the Encoding team maps it onto the factored problem tree𝒯P\\mathcal\{T\}\_\{P\}, returning the fingerprintpjp\_\{j\}\. \(D\) Warm\-start: the nearest neighbors ofpjp\_\{j\}in𝒟\\mathcal\{D\}, weighted by fingerprint similarity and reward and blended with a uniform prior, induce a probabilistic action space on𝒯A\\mathcal\{T\}\_\{A\}; the same neighbors supply expected observables anchoring realistic targets\. \(E\) Per\-trial loop: the Strategy team \(proposer plus critic\) samples a fingerprintmim\_\{i\}from𝒯A\\mathcal\{T\}\_\{A\}along GRAFT’s decision levels; the Implementation team realizesmim\_\{i\}as runnable codeSiS\_\{i\}and returns observablesOiO\_\{i\}; the Advisor team \(diagnosticians plus advisor agents\) scores the run, computesrir\_\{i\}, and on failure either revises the proposal using per\-node general hints or extends a branch with a new node \(𝒢A\\mathcal\{G\}\_\{A\}grows mid\-trial\); on success it also writes new hints onto the touched nodes\. Every run, success or failure, commits to𝒟\\mathcal\{D\}\. The legend identifies the LLM driving each team\.
2. 2\.GRAFT\-ATHENA couples this substrate to dual problem and action knowledge graphs𝒢P\\mathcal\{G\}\_\{P\}and𝒢A\\mathcal\{G\}\_\{A\}, persistent memory𝒟\\mathcal\{D\}, formalization and well\-posedness teams, autonomous solver\-ingestion teams, and a closed\-loop implementation/advisor cycle that lets the system learn across problems \(§[2\.1](https://arxiv.org/html/2605.11117#S2.SS1), Methods §[4\.2](https://arxiv.org/html/2605.11117#S4.SS2), §[4\.1\.8](https://arxiv.org/html/2605.11117#S4.SS1.SSS8)\)\.
3. 3\.The same machinery produces concrete scientific artefacts, including production\-solver configurations on Trixi\.jl and Nektar\+\+ \(§[2\.5](https://arxiv.org/html/2605.11117#S2.SS5), Appendix[C](https://arxiv.org/html/2605.11117#A3)\), particle\-based DPD studies of red\-blood\-cell rheology \(§[2\.6](https://arxiv.org/html/2605.11117#S2.SS6)\), well\-posed reformulations of in\-vivo inverse problems with new regularizers \(§[2\.7](https://arxiv.org/html/2605.11117#S2.SS7)\), and an agent\-designed spectral PINN with exponential convergence \(§[2\.8](https://arxiv.org/html/2605.11117#S2.SS8)\); every artefact persists in𝒟\\mathcal\{D\}for future runs\.

Considered together, GRAFT\-ATHENA gives autonomous scientific computing a geometric memory: a joint problem\-method space in which evaluated observables such as reward, relativeL2L^\{2\}error, wall time, and system size are attached to recorded\(p,m\)\(p,m\)pairs rather than isolated runs\. As this space fills, search, calibration, and eventually counterfactual reasoning become operations on a navigable geometry instead of bespoke acts of problem solving\.

## 2Results

### 2\.1GRAFT\-ATHENA: a self\-improving agentic framework built on probabilistic decision trees\.

In this study, we present GRAFT\-ATHENA, an agentic framework for scientific computing and scientific machine learning that learns as it solves more problems, ingests new software and methods on its own, and self\-calibrates as new evidence arrives\. Formally, we model GRAFT\-ATHENA as a stochastic policy that, given a problemp∈𝒫p\\in\\mathcal\{P\}, places a distribution over candidate methodsm∈ℳm\\in\\mathcal\{M\}rather than picking one deterministically; each method is itself a tuple of actionsm=\(a1,…,ak\)m=\(a\_\{1\},\\dots,a\_\{k\}\)drawn from an action vocabulary𝒜\\mathcal\{A\}\(Methods, §[4\.1\.1](https://arxiv.org/html/2605.11117#S4.SS1.SSS1)\)\. Many solver\-facing scientific\-computing workflows expose a finite configuration layer that can be represented as a directed acyclic graph \(DAG\), since each is built from attributes whose options interact through dependencies\. We exploit this observation to introduce GRAFT \(Graph Reduction to Adaptive Factored Trees\), a construction that converts the problem and action DAGs into factored trees that enumerate every attribute and every option in the domain\. Each problem and each method then corresponds to a single path through its tree, and the signature of that path is its fingerprint\. Crucially, GRAFT confers three abilities that the rest of this section unpacks\. First, GRAFT\-ATHENA learns from previously solved problems by transporting methods that succeeded on fingerprint\-similar problems, so the action space carries the experience of every past run\. Second, it grows its own knowledge graph by absorbing new software and methods directly from the documentation of production solvers, so the substrate itself enlarges as new solvers are encountered\. Third, it self\-calibrates expected outcomes from the performance recorded on neighboring problems, anchoring realistic targets for every new run\.

Fig\.[1](https://arxiv.org/html/2605.11117#S1.F1)A shows the framework’s capability to extend its own knowledge graph\. An Expansion team draws on the online documentation of a production solver, the published methods literature, or input from a domain expert, and emits two structured artefacts: a draft DAG of the method’s attributes and a set of hints extracted from the source\. A Construction team then integrates the draft into the existing knowledge graph, classifying each hint as either a cross\-attribute dependency \(which adds a new connection between attributes\) or a general hint \(which decorates the relevant node\), in a back\-and\-forth with GRAFT \(§[2\.2](https://arxiv.org/html/2605.11117#S2.SS2)\) that rejects fragments introducing cycles or violating the hint schema, and revises until the addition admits a rule\-preserving tree projection\. Fig\.[1](https://arxiv.org/html/2605.11117#S1.F1)B shows two complementary objects\. The structural substrate is given by two knowledge graphs:𝒢P\\mathcal\{G\}\_\{P\}catalogues the problem space \(PDE and DPD problems in this study\), and𝒢A\\mathcal\{G\}\_\{A\}the action vocabulary of the method space \(PIMLToscanoet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib23)\], classical numerical solvers, and DPDThompsonet al\.\[[2022](https://arxiv.org/html/2605.11117#bib.bib63)\]schemes; the numerical family covers Trixi\.jlRanochaet al\.\[[2021](https://arxiv.org/html/2605.11117#bib.bib220)\], Nektar\+\+Cantwellet al\.\[[2015](https://arxiv.org/html/2605.11117#bib.bib14)\], and pseudospectral methods in Julia\)\. The long\-term memory𝒟=\{\(pj,mi,Oi,ri\)\}\\mathcal\{D\}=\\\{\(p\_\{j\},m\_\{i\},O\_\{i\},r\_\{i\}\)\\\}then records each solved instance on this substrate, withpjp\_\{j\}a path through𝒢P\\mathcal\{G\}\_\{P\},mim\_\{i\}a path through𝒢A\\mathcal\{G\}\_\{A\},OiO\_\{i\}a tuple of run\-time observables \(e\.g\., wall\-clock timetit\_\{i\}, relativeL2L^\{2\}erroreie\_\{i\}, number of atoms, etc\.\), andri∈\[0,rmax\]r\_\{i\}\\in\[0,r\_\{\\max\}\]a scalar reward derived fromOiO\_\{i\}that grades how wellmim\_\{i\}performed\. Notice that this construction allows the same problempjp\_\{j\}to be solved by multiple methodsmim\_\{i\}, each with a potentially different reward\. The graph itself can grow: when the Expansion and Construction teams add new nodes to𝒢P\\mathcal\{G\}\_\{P\}or𝒢A\\mathcal\{G\}\_\{A\}, every past entry in𝒟\\mathcal\{D\}stays valid, since its recorded paths were taken on the smaller graph and do not reference the new nodes\.

Fig\.[1](https://arxiv.org/html/2605.11117#S1.F1)C shows the static setup that brings a new user problem into the system\. The Formalization team turns the user’s free\-form problem statement into a structured user request: it extracts the governing equations, boundary conditions, and data; reduces the problem where possible via symmetries, coordinate transforms, or exact solutions; and audits well\-posedness, optionally adding preprocessing steps or extra constraints\. Throughout, the team is guided by an analytical DAG𝒢F\\mathcal\{G\}\_\{F\}of simplification moves \(symmetry reductions, coordinate transforms, non\-dimensionalizations, candidate exact solutions\), which GRAFT projects into a factored analytical tree𝒯F\\mathcal\{T\}\_\{F\}that the agents read as scaffolding; the tree informs without constraining, since analytical reasoning is more open\-ended than the action vocabulary of𝒯A\\mathcal\{T\}\_\{A\}\. The Encoding team then maps the user request onto the problem tree: GRAFT is invoked at this point to project the knowledge graph𝒢P\\mathcal\{G\}\_\{P\}into the factored tree𝒯P\\mathcal\{T\}\_\{P\}and its decision levels \(§[2\.2](https://arxiv.org/html/2605.11117#S2.SS2)\), and the team encodes the request as a path through𝒯P\\mathcal\{T\}\_\{P\}, whose signature is the problem fingerprintpjp\_\{j\}\. Fig\.[1](https://arxiv.org/html/2605.11117#S1.F1)D shows how this fingerprint warm\-starts an action space tailored topjp\_\{j\}\. GRAFT places every problem fingerprint at a deterministic location in a shared geometry, so distances between fingerprints are well defined and the nearest neighbors ofpjp\_\{j\}in𝒟\\mathcal\{D\}can be retrieved directly\. Each retrieved neighbor\(pi,mi,Oi,ri\)\(p\_\{i\},m\_\{i\},O\_\{i\},r\_\{i\}\)contributes its methodmim\_\{i\}to a blend weighted by fingerprint similarity and reward and combined with a uniform prior \(Methods, §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)\), and the result is a probabilistic action space: the same action tree𝒯A\\mathcal\{T\}\_\{A\}, but with per\-node probabilities biased towards the actions that worked on fingerprint\-similar past problems\. The same neighborhood also fixes realistic targets for the new run, since each neighbor’s observables \(wall time, error floor, system size\) and rewardrir\_\{i\}together calibrate what a successful run onpjp\_\{j\}should achieve, anchoring the reward signal that will drive the loop of panel E\.

Fig\.[1](https://arxiv.org/html/2605.11117#S1.F1)E shows the per\-trial loop that traverses this action space\. The Strategy team \(proposer plus critic\) navigates𝒯A\\mathcal\{T\}\_\{A\}along GRAFT’s decision levels and emits a proposed method fingerprintmim\_\{i\}\. The Implementation team realizesmim\_\{i\}as runnable codeSiS\_\{i\}, handles preprocessing \(e\.g\., mesh generation, data analysis\), debugging, monitoring, postprocessing \(e\.g\., image generation, metrics extraction\) and returns the observablesOiO\_\{i\}\. The Advisor team \(diagnosticians plus advisor agents\) reads the executed path andOiO\_\{i\}, scores along accuracy, efficiency and other metrics, and computes the rewardrir\_\{i\}\. Every run is committed to𝒟\\mathcal\{D\}as\(pj,mi,Oi,ri\)\(p\_\{j\},m\_\{i\},O\_\{i\},r\_\{i\}\), so low\-reward attempts discount their actions in future priors\. On failure, the Advisor revises the proposal itself: it localizes the failure to a decision level on𝒯A\\mathcal\{T\}\_\{A\}, validates the revision against GRAFT, and consults the per\-node general hints \(e\.g\., periodic embeddings for periodic boundary conditions\); or it extends a branch with a new node \(e\.g\., a deeper architecture, a different embedding\), in which case𝒢A\\mathcal\{G\}\_\{A\}grows and𝒯A\\mathcal\{T\}\_\{A\}regrows before the next attempt; the revised method goes to Implementation and the cycle repeats\. A short\-term trial history𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}records every attempted\(m,O,r\)\(m,O,r\)onpjp\_\{j\}, so the Advisor never reissues a failed configuration; at trial end it folds into𝒟\\mathcal\{D\}\(Methods, §[4\.1\.8](https://arxiv.org/html/2605.11117#S4.SS1.SSS8)\)\. On success, the Advisor also writes new general hints onto the touched nodes, populating the same store Expansion seeded in panel A; the hint layer is co\-authored by experts and by GRAFT\-ATHENA\. Self\-improvement runs on two axes: cross\-problem via𝒟\\mathcal\{D\}’s prior on𝒯A\\mathcal\{T\}\_\{A\}, and within\-problem via the Advisor’s directed correction and the growing hint store\.

### 2\.2GRAFT: from a combinatorial graph to a navigable substrate

![Refer to caption](https://arxiv.org/html/2605.11117v1/Figures/GRAFT.png)Figure 2:From example domain to production landscape: the GRAFT encoding pipeline\.\(A\) Example DAG: greencc\-edges, bluess\-edges, and a red cross\-edge for forced combinations\. Three chains hang off morning \(breakfast, clothes, transport\); clothes splits via two furthercc\-edges into style and helmet sub\-chains, and the cross\-rule pins helmet to yes whenever bike is picked\. \(B\) Decision\-level projection: rule\-free chains at level 1; helmet drops to level 2, conditioned on transport, with the cross\-rule the one red edge crossing the boundary\. \(C\) Production DAG covering all problem and method spaces\. \(D\) Tree of \(B\) embedded in\[0,1\]3\[0,1\]^\{3\}by the partition\-of\-unity layout, with methodM1M\_\{1\}traced through the cube and its fingerprint shadow on the\(x,y\)\(x,y\)floor; the floor also carries the fingerprint shadows of alternative methodsM2M\_\{2\}andM3M\_\{3\}, withM1M\_\{1\}closer toM3M\_\{3\}than toM2M\_\{2\}by eye\. \(E\) HasegawaWakatani2D’s path through the production problem tree with fingerprint shadow\. \(F\) The Nektar configuration paired with it on the methods tree\. \(G\) Landscape view: fingerprints reduced to one PCA coordinate per axis \(problems onxx, methods onyy\), one subplot per family, colored by mean relativeL2L^\{2\}error \(PIML\), wall time \(Numerical\), and system sizenatomsn\_\{\\text\{atoms\}\}\(DPD\)\. As more problems are solved the landscape fills in and ATHENA learns the problems\-to\-methods map; the per\-family metrics anchor reference observables that qualify future problems via their closest neighbors\.Solving a problem in the real world often involves probabilistic factors: features of the problem tilt the relative likelihood of one method over another, and a handful of hard constraints can force certain method components together\. Consider, for instance, planning a morning routine, which depends on several factors such as day of the week, rain, meetings, etc\. Notice that these factors shape the choice of method \(e\.g\., a rainy day tilts the transport choice towards the car\), while others, in turn, force a selection outright \(e\.g\., deciding to ride a bike requires wearing a helmet\)\. To model this probabilistic structure, we encode the problem and the method as a probabilistic directed acyclic graph, identifying three kinds of relations: attributes that apply jointly \(e\.g\., morning has a breakfast \(𝒜1\\mathcal\{A\}\_\{1\}\), a clothes, and a transport \(𝒜2\\mathcal\{A\}\_\{2\}\) attribute, with the clothes attribute itself bundling a style \(𝒜3\\mathcal\{A\}\_\{3\}\) and a helmet \(𝒜4\\mathcal\{A\}\_\{4\}\) sub\-attribute\), single decisions on each chainjjthat resolve by picking one option from a mutually exclusive alphabet𝒜j\\mathcal\{A\}\_\{j\}\(e\.g\., transport bike or car, i\.e\.,\|𝒜2\|=2\|\\mathcal\{A\}\_\{2\}\|=2\), and dependencies between chains that fold in the hard or soft cross\-rules \(e\.g\., bike requires helmet\) \(see Fig\.[2](https://arxiv.org/html/2605.11117#S2.F2)A\)\. The same construction applies to the problem itself since its descriptors \(e\.g\., weekend/weekday, raining, meeting\) become chains of a problem\-side DAG\. Therefore, in this setup, an intelligent system can be described as a map from one such graph to the other \(i\.e\., given a draw from the problem side, return a draw from the method side\)\. A direct way to encode that map is to flatten each DAG into a single categorical over its joint configuration; the formulation is correct but already costs, for this binary case,\|ℳ0\|=∏j=1k\|𝒜j\|=16\|\\mathcal\{M\}\_\{0\}\|=\\prod\_\{j=1\}^\{k\}\|\\mathcal\{A\}\_\{j\}\|=16entries for the morning example, and the count grows exponentially in both the number of chains and the number of conditioning relations\. For instance, the production DAG in Fig\.[2](https://arxiv.org/html/2605.11117#S2.F2)C has thousands of nodes across dozens of chains, sending\|ℳp\|\|\\mathcal\{M\}\_\{p\}\|well past anything a policy could realistically learn\.

Even though this count seems computationally prohibitive, notice that most of these decisions can be taken independently: whether to have breakfast, for instance, does not affect whether to wear a helmet, and the choice of style does not constrain the choice of transport\. It is therefore possible to split the DAG into independent Markov decision processes, one per chain, which reduces the joint from exponential to linear in the number of chains \(i\.e\.,\|ℳ0\|=∏j=1k\|𝒜j\|\|\\mathcal\{M\}\_\{0\}\|=\\prod\_\{j=1\}^\{k\}\|\\mathcal\{A\}\_\{j\}\|to∑j=1k\|𝒜j\|\\sum\_\{j=1\}^\{k\}\|\\mathcal\{A\}\_\{j\}\|\)\. The cross\-rules cannot simply be dropped, however, since they record genuine couplings between otherwise independent chains; we reintroduce them as preconditions on the affected chain alone, firing locally once their trigger has been resolved upstream \(e\.g\., the transport chain preconditions the helmet chain as shown in Fig\.[2](https://arxiv.org/html/2605.11117#S2.F2)B\)\. This naturally gives rise to decision levels: chains whose triggers reference no other chain sit at level 1 and are sampled in parallel, while chains conditioned on a lower\-level value sit at the next level and are sampled sequentially after their triggers resolve\. The resulting representation, in the lineage of the Bayesian\-network factorizations introduced in the late 1980sPearl \[[1988](https://arxiv.org/html/2605.11117#bib.bib781)\], Verma and Pearl \[[1988](https://arxiv.org/html/2605.11117#bib.bib782)\], Lauritzen and Spiegelhalter \[[1988](https://arxiv.org/html/2605.11117#bib.bib809)\], Verma and Pearl \[[1990](https://arxiv.org/html/2605.11117#bib.bib804)\], has a parameter footprint of∑j=1k\|𝒜j\|\+\|ℛ\|\\sum\_\{j=1\}^\{k\}\|\\mathcal\{A\}\_\{j\}\|\+\|\\mathcal\{R\}\|, scaling linearly in the number of chains rather than exponentially in the number of decisions\. Crucially, no encoded cross\-rule is dropped: under mild locality assumptions on the proposer–critic pair \(Assumption[4\.3](https://arxiv.org/html/2605.11117#S4.Thmtheorem3)\), every dependency in the original DAG is preserved as an explicit cross\-rule rather than baked into the joint, with the formalII\-map argument and proof deferred to Methods \(Proposition[4\.1](https://arxiv.org/html/2605.11117#S4.Thmtheorem1), Appendix[A\.1](https://arxiv.org/html/2605.11117#A1.SS1)\)\.

Notice that this construction is general: problems from different fields slot into the same setup just by enumerating their descriptors on the problem side and their candidate solution components on the method side\. We call this construction GRAFT \(Graph Reduction to Adaptive Factored Trees\), and use the term throughout for the substrate it produces: the factored trees𝒯P\\mathcal\{T\}\_\{P\}and𝒯A\\mathcal\{T\}\_\{A\}on the problem and method sides, their decision levels, the rule setℛ\\mathcal\{R\}that retains the cross\-couplings dropped by the spanning\-tree projection, and the embedding introduced below\.

In this study we instantiate the problem space as the union of partial\-differential\-equation \(PDE\) problems and dissipative\-particle\-dynamics \(DPD\) problems, and the method space as their candidate solution strategies, including physics\-informed machine learning \(PIML\), classical numerical solvers, and DPD schemes\. To exploit this generality, we deploy the agentic exploration team to aid the construction of the graph: the tree structure is lifted directly from each solver’s code documentation, and the cross\-rules are extracted from the surrounding documentation prose\. End\-to\-end on the two production frameworks used here, this yields a Trixi\.jl subtree with176176nodes and5151cross\-rules and a Nektar\+\+ subtree with394394nodes and6060cross\-rules, contributing111111documented dependencies to the cross\-rule storeℛ\\mathcal\{R\}for the Numerical family\. The combined problem and method graphs, with the Trixi\.jl and Nektar\+\+ subtrees built end\-to\-end by the agents and the PIML and DPD subtrees user\-defined, are shown in Fig\.[2](https://arxiv.org/html/2605.11117#S2.F2)C; the agent\-built numerical\-methods slice of𝒯A\\mathcal\{T\}\_\{A\}is reproduced as a nested decision\-chain listing in Appendix[C](https://arxiv.org/html/2605.11117#A3)as the reproducible artifact backing the numerical\-side runs\. Each problem solved on these trees, together with the method that solved it and the score it earned, is committed to𝒟\\mathcal\{D\}, ATHENA’s long\-term memory: a persistent store that grows with every solved problem and is consulted on every new arrival\.

### 2\.3Unique fingerprints, computable distances, self\-calibrating rewards

In this study, we define ATHENA as a map between problems and solutions: given a problem, it returns a method to solve it\. More precisely, the system carries a policyπ\(⋅∣p\)\\pi\(\\cdot\\mid p\)that, for each problempp, places a distribution over candidate methods \(Methods §[4\.1\.1](https://arxiv.org/html/2605.11117#S4.SS1.SSS1)\)\. GRAFT gives this map the right structure to act on, since each problem is a path through𝒯P\\mathcal\{T\}\_\{P\}, each method a path through𝒯A\\mathcal\{T\}\_\{A\}, and the cross\-rules inℛ\\mathcal\{R\}carry the couplings the spanning\-tree projection cannot\. Ideally, ATHENA should improve as it gains experience, reusing older problems to solve new ones; this requires a notion of proximity between paths, which the structure itself does not supply\. To address this, we propose a partition\-of\-unity layoutΦ\\Phithat places every node at a deterministic position in the unit cube \(Algorithm[3](https://arxiv.org/html/2605.11117#alg3), Appendix[A\.2](https://arxiv.org/html/2605.11117#A1.SS2)\), and show that this layout is injective \(Proposition[4\.2](https://arxiv.org/html/2605.11117#S4.Thmtheorem2)\)\. A subsequent projection of the cube onto the\(x,y\)\(x,y\)floor, followed by discretization at sufficient resolution \(Appendix[A\.2](https://arxiv.org/html/2605.11117#A1.SS2)\), assigns each node to its own cell\. We then define the fingerprint of a method \(or problem\) as the set of cells visited by its path through the corresponding tree \(Eq\. \([10](https://arxiv.org/html/2605.11117#S4.E10)\)\), which by construction uniquely identifies it\. Fig\.[2](https://arxiv.org/html/2605.11117#S2.F2)D illustrates the construction on the example tree, with one methodM1M\_\{1\}traced through the cube and its fingerprint shadow on the\(x,y\)\(x,y\)floor; the same floor carries the fingerprint shadows of two alternative methodsM2M\_\{2\}andM3M\_\{3\}, and proximity already reads as similarity by eye, withM1M\_\{1\}closer toM3M\_\{3\}than toM2M\_\{2\}\. The same construction applies, without modification, to the production problem and method trees: Fig\.[2](https://arxiv.org/html/2605.11117#S2.F2)E traces HasegawaWakatani2D’s path through the production problem tree with its fingerprint shadow, and Fig\.[2](https://arxiv.org/html/2605.11117#S2.F2)F the paired Nektar configuration on the Numerical methods tree\.

Notice that what panels E and F have just shown is exactly the kind of object ATHENA is trying to produce: a problem paired with a method that solves it\. Because both sides now live in the same kind of space \(paths embedded byΦ\\Phiinto the unit cube\), it is natural to consider functions on this joint space, such as errors, wall times, system sizes, or even images of the produced solutions\. Evaluating such functions on a solved instance\(pi,mi\)\(p\_\{i\},m\_\{i\}\)yields the observableOiO\_\{i\}, and a further function ofOiO\_\{i\}defines the scalar rewardri∈\[0,rmax\]r\_\{i\}\\in\[0,r\_\{\\max\}\]that grades the solution \(Methods, §[4\.1\.1](https://arxiv.org/html/2605.11117#S4.SS1.SSS1)\); the long\-term memory𝒟\\mathcal\{D\}is then the collection of these input\-output tuples\{\(pi,mi,Oi,ri\)\}\\\{\(p\_\{i\},m\_\{i\},O\_\{i\},r\_\{i\}\)\\\}\. To visualize this, Fig\.[2](https://arxiv.org/html/2605.11117#S2.F2)G reduces each fingerprint to one PCA coordinate per axis \(problems onxx, methods onyy\) and colors each solved instance by one such function: relativeL2L^\{2\}erroreie\_\{i\}for PIML, wall timetit\_\{i\}for Numerical, system sizenatomsn\_\{\\mathrm\{atoms\}\}for DPD\. Since both problems and methods now live in this geometry, an incoming problem can be measured against the entries already in𝒟\\mathcal\{D\}: its closest neighbors give a starting point for exploration, and their attached observables and rewards calibrate realistic targets \(a neighbor solved in one hour predicts a one\-hour budget; one converged at10−810^\{\-8\}predicts an attainable error floor\)\. As\|𝒟\|\|\\mathcal\{D\}\|grows the landscape of Fig\.[2](https://arxiv.org/html/2605.11117#S2.F2)G fills in, and the substrate matures into a navigable atlas of methods indexed by the problems they solve\. The same neighbors also signal which methods are likely to succeed on the incoming problem: across the PIML runs of §[2\.4](https://arxiv.org/html/2605.11117#S2.SS4), the agent identified the combination of Self\-Scaled Broyden with backtrackingJniniet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib11)\]line search as a high\-reward optimizer choice from past trials in𝒟\\mathcal\{D\}, and the prior carried it forward to every subsequent run in the family\. Picking the single nearest neighbor, however, would be deterministic, while ATHENA needs room to explore; we therefore take many neighbors and combine them into a probabilistic map from problems to methods\. As𝒟\\mathcal\{D\}grows, this map biases new problems towards the methods that solved their nearest past ones, letting ATHENA remember what worked and learn from past experience; the construction of this map is the subject of the next section\.

![Refer to caption](https://arxiv.org/html/2605.11117v1/Figures/Priors_and_results.png)Figure 3:Self\-updating priors and converged solutions across four PDE benchmarks\.\(A\) Target fingerprint of Viscous Burgers \(ν=1/\(100​π\)\\nu=1/\(100\\pi\)\) on𝒯P\\mathcal\{T\}\_\{P\}, displayed at the coarse visualization resolutionK=32K=32\(Eq\.[10](https://arxiv.org/html/2605.11117#S4.E10)\), with neighbor ranking on𝒟\\mathcal\{D\}performed at the identity\-preserving resolutionK⋆K^\{\\star\}\(Methods, §[4\.1\.5](https://arxiv.org/html/2605.11117#S4.SS1.SSS5)\); black cells mark its root\-to\-leaf path\. \(B\) Three nearest past problems in𝒟\\mathcal\{D\}underJK⋆J\_\{K^\{\\star\}\}\(J=\{0\.60,0\.45,0\.44\}J=\\\{0\.60,\\,0\.45,\\,0\.44\\\}, top row\), paired column\-wise with the methods on𝒯A\\mathcal\{T\}\_\{A\}that solved them \(4949,4444,4545active cells, bottom row\); each pair is a partial specification feeding the blend\. \(C\) Merged action prior on𝒯A\\mathcal\{T\}\_\{A\}from the sigmoid\-gated, reward\-weighted blend \(Eqs\.[12](https://arxiv.org/html/2605.11117#S4.E12)–[15](https://arxiv.org/html/2605.11117#S4.E15)\); cell shade is the probability mass accumulated on the corresponding child of𝒯A\\mathcal\{T\}\_\{A\}, read off the shared colorbar \(right\) in\[0,1\]\[0,1\], with164164cells carrying non\-zero support and the darker cells marking the children selected by the closest, highest\-reward neighbors\. \(D\) Converged solutions on Viscous Burgers 1D, KdV, and Helmholtz 2D\. Columns: predicted fielduθu\_\{\\theta\};\|uθ−uref\|\|u\_\{\\theta\}\-u\_\{\\mathrm\{ref\}\}\|\(log\); PDE residual \(log\); relativeL2L^\{2\}vs iteration, with GRAFT\-ATHENA \(red\) compared with ATHENA \(blue\) as a recent agentic baseline\. GRAFT\-ATHENA reaches a lower error floor on all three; the Viscous Burgers trace shows the≈25,000\\approx 25\{,\}000\-iteration plateau and sharp decrease characteristic of the inherited Reynolds\-number continuation\.
### 2\.4Self\-updating priors: how GRAFT\-ATHENA learns from experience

Table 1:Quantitative comparison against state\-of\-the\-art baselines\.We evaluate GRAFT\-ATHENA across four canonical PIML benchmarks against recent expert\-authored publications \(Human\) and agentic frameworks, including ATHENA\. The primary accuracy metric is the relativeL2L^\{2\}errorR​L2RL\_\{2\}\(Eq\. \([18](https://arxiv.org/html/2605.11117#S4.E18)\)\) on a fixed dense evaluation grid; MSE is reported as a secondary diagnostic, namely the mean squared PDE residual on the same grid\. Because residual MSE and solution\-field error are not equivalent objectives in PINNs, methods are ranked here primarily byR​L2RL\_\{2\}\. ATHENA reference values are recomputed from its public run artifacts \(min\\minover the training history\) so the comparison shares methodology with the GRAFT\-ATHENA reports; cited values from other baselines are quoted as published rather than recomputed, and dashes indicate metrics not reported in the cited source\.ProblemReferenceBenchmark TypeRel\.L2L^\{2\}ErrorMSEBurgers \(ν=1100​π\\nu=\\frac\{1\}\{100\\pi\}\)\[Wanget al\.,[2025a](https://arxiv.org/html/2605.11117#bib.bib44)\]Human4\.03×10−54\.03\\times 10^\{\-5\}–\[Chenet al\.,[2025](https://arxiv.org/html/2605.11117#bib.bib51)\]Human1\.38×10−41\.38\\times 10^\{\-4\}–\[Wuwuet al\.,[2025](https://arxiv.org/html/2605.11117#bib.bib35)\]Agents–6\.51×10−56\.51\\times 10^\{\-5\}\[Heet al\.,[2025](https://arxiv.org/html/2605.11117#bib.bib26)\]Agents–6\.48×10−56\.48\\times 10^\{\-5\}\[Urbánet al\.,[2024](https://arxiv.org/html/2605.11117#bib.bib496)\]Human2\.90×10−62\.90\\times 10^\{\-6\}–\[Kiyaniet al\.,[2025](https://arxiv.org/html/2605.11117#bib.bib38)\]Human1\.62×10−81\.62\\times 10^\{\-8\}–Toscanoet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib23)\]Human8\.25×10−98\.25\\times 10^\{\-9\}–\[Toscanoet al\.,[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\]Agents7\.88×10−97\.88\\times 10^\{\-9\}4\.33×10−144\.33\\times 10^\{\-14\}OursAgents1\.48×𝟏𝟎−𝟗\\mathbf\{1\.48\\times 10^\{\-9\}\}7\.64×𝟏𝟎−𝟏𝟔\\mathbf\{7\.64\\times 10^\{\-16\}\}KdV\[Urbánet al\.,[2024](https://arxiv.org/html/2605.11117#bib.bib496)\]Human6\.00×10−66\.00\\times 10^\{\-6\}–\[Toscanoet al\.,[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\]Agents7\.92×10−87\.92\\times 10^\{\-8\}1\.22×10−131\.22\\times 10^\{\-13\}OursAgents4\.38×𝟏𝟎−𝟖\\mathbf\{4\.38\\times 10^\{\-8\}\}4\.32×𝟏𝟎−𝟏𝟓\\mathbf\{4\.32\\times 10^\{\-15\}\}Helmholtz\[Urbánet al\.,[2024](https://arxiv.org/html/2605.11117#bib.bib496)\]Human3\.60×10−73\.60\\times 10^\{\-7\}–\[Chenet al\.,[2025](https://arxiv.org/html/2605.11117#bib.bib51)\]Human4\.86×10−54\.86\\times 10^\{\-5\}–Jniniet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib11)\]Human2\.0×10−92\.0\\times 10^\{\-9\}–\[Toscanoet al\.,[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\]Agents7\.25×10−97\.25\\times 10^\{\-9\}7\.25×10−137\.25\\times 10^\{\-13\}OursAgents6\.20×𝟏𝟎−𝟏𝟎\\mathbf\{6\.20\\times 10^\{\-10\}\}4\.65×𝟏𝟎−𝟏𝟓\\mathbf\{4\.65\\times 10^\{\-15\}\}Poisson\[Wanget al\.,[2025a](https://arxiv.org/html/2605.11117#bib.bib44)\]Human2\.99×10−72\.99\\times 10^\{\-7\}–\[Müller and Zeinhofer,[2023](https://arxiv.org/html/2605.11117#bib.bib492)\]Human1\.0×10−71\.0\\times 10^\{\-7\}–\[Toscanoet al\.,[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\]Agents8\.13×10−98\.13\\times 10^\{\-9\}3\.95×𝟏𝟎−𝟏𝟑\\mathbf\{3\.95\\times 10^\{\-13\}\}OursAgents5\.05×𝟏𝟎−𝟗\\mathbf\{5\.05\\times 10^\{\-9\}\}4\.51×10−134\.51\\times 10^\{\-13\}The fingerprint construction of the previous subsection assigns each path through𝒯P\\mathcal\{T\}\_\{P\}and𝒯A\\mathcal\{T\}\_\{A\}a unique set of cells, so proximity between two of them reduces to set comparison\. We adopt the Jaccard distancedJ​\(Fi,Fj\)=1−\|Fi∩Fj\|/\|Fi∪Fj\|d\_\{J\}\(F\_\{i\},F\_\{j\}\)=1\-\|F\_\{i\}\\cap F\_\{j\}\|/\|F\_\{i\}\\cup F\_\{j\}\|on the resulting fingerprints \(Eq\.[11](https://arxiv.org/html/2605.11117#S4.E11)\), a true metric on non\-empty finite sets \(Proposition[A\.3](https://arxiv.org/html/2605.11117#A1.Thmtheorem3), Appendix[A\.3](https://arxiv.org/html/2605.11117#A1.SS3)\)\. Equipped with this distance, the goal is to bias the action prior on𝒯A\\mathcal\{T\}\_\{A\}towards methods that succeeded on problems close to the incomingpnewp\_\{\\text\{new\}\}, illustrated throughout this section on the Viscous Burgers equation, whose target fingerprint on𝒯P\\mathcal\{T\}\_\{P\}is shown in Fig\.[3](https://arxiv.org/html/2605.11117#S2.F3)A\. Specifically, for each solved instance\(pi,mi,Oi,ri\)∈𝒟\(p\_\{i\},m\_\{i\},O\_\{i\},r\_\{i\}\)\\in\\mathcal\{D\}we define a confidence weightwiw\_\{i\}that grows with the fingerprint similarity betweenpip\_\{i\}andpnewp\_\{\\text\{new\}\}and with the recorded rewardri/rmaxr\_\{i\}/r\_\{\\max\}, so a neighbor contributes to the prior only when it is both close and successful \(Eq\.[12](https://arxiv.org/html/2605.11117#S4.E12), Methods §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)\)\. Notice that eachmim\_\{i\}is itself a path through𝒯A\\mathcal\{T\}\_\{A\}and therefore already encodes, at every internal node it visits, which child the neighbor selected; in this sense each neighbor is a partial specification of the action prior\. Fig\.[3](https://arxiv.org/html/2605.11117#S2.F3)B shows the three closest neighbors of Viscous Burgers in𝒟\\mathcal\{D\}underdJd\_\{J\}on the top row, paired with the method fingerprints on𝒯A\\mathcal\{T\}\_\{A\}they were solved with on the bottom row; these three pairs supply the partial specifications that feed the blend\. We aggregate them into a single prior on𝒯A\\mathcal\{T\}\_\{A\}by averaging their row contributions weighted bywiw\_\{i\}, and blend the result with a uniform prior on the rows where no neighbor has voted \(Eq\.[15](https://arxiv.org/html/2605.11117#S4.E15)\)\. The resulting prior fingerprint is shown in Fig\.[3](https://arxiv.org/html/2605.11117#S2.F3)C, where the darker cells correspond to the children that accumulated the most mass in the blend, namely the actions selected by the closest, highest\-reward neighbors; the proposer of the next stage will sample from this prior, biased, by construction, towards actions that have already worked on comparable problems\. The closed\-form update is given in Methods \(§[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)\)\.

With the prior in place, the per\-trial loop of §[2\.1](https://arxiv.org/html/2605.11117#S2.SS1)samples a method, runs it, and folds the outcome back into𝒟\\mathcal\{D\}\(Methods, §[4\.1\.8](https://arxiv.org/html/2605.11117#S4.SS1.SSS8)\)\. Fig\.[3](https://arxiv.org/html/2605.11117#S2.F3)D reports the resulting relativeL2L^\{2\}error against iteration count on three representative cases \(Viscous Burgers atν=1/\(100​π\)\\nu=1/\(100\\pi\), KdV, and Helmholtz 2D\), and Table[1](https://arxiv.org/html/2605.11117#S2.T1)compares the converged accuracy against expert\-authored baselines and recent agentic systems, including ATHENAToscanoet al\.\[[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\], across four canonical PIML benchmarks; code to reproduce the runs is released upon publication\. Across all four, GRAFT\-ATHENA reaches a lower error floor than this earlier agentic baseline\. The mechanism is most cleanly traced on Viscous Burgers: the ATHENA trace stabilizes near10−810^\{\-8\}while GRAFT\-ATHENA descends to∼10−9\\sim 10^\{\-9\}, with the residual trace showing a flat early plateau over the first≈25,000\\approx 25\{,\}000iterations followed by a sharp decrease, the signature of a Reynolds\-number continuation that the prior pulled in from past higher\-Re\\mathrm\{Re\}runs and applied here even though the targetRe\\mathrm\{Re\}is moderate\. The remaining cases in Table[1](https://arxiv.org/html/2605.11117#S2.T1)span structurally distinct PDE families: KdV is third\-order dispersive with mixed Neumann and Dirichlet conditions, Helmholtz 2D is elliptic with periodic boundaries, and Poisson is elliptic with Dirichlet conditions; across all four, the converged MSE lies between10−1610^\{\-16\}and10−1310^\{\-13\}, bordering double\-precision machine arithmetic\.

### 2\.5From schematic to shockwave: autonomous solution of the Apollo capsule at Mach 10\.

We pose the system the simulation of an Apollo command module re\-entering the atmosphere atMa=10\\mathrm\{Ma\}=10, the configuration documented in NASA’s 1968 postflight aerodynamics reportGriffith and Boylan \[[1968](https://arxiv.org/html/2605.11117#bib.bib16)\]\(Fig\.[4](https://arxiv.org/html/2605.11117#S2.F4)A\)\. The case is a textbook stress test for hypersonic numerics\. A strong bow shock stands off the heatshield \(Fig\.[4](https://arxiv.org/html/2605.11117#S2.F4)D, top row\), and behind that shock the gas decelerates to near zero velocity at the stagnation point on the body’s nose, where pressure and density swing across orders of magnitude inside a region a few cells wide \(Fig\.[4](https://arxiv.org/html/2605.11117#S2.F4)D, middle row, where the local Mach number collapses into the dark\-blue subsonic pocket\)\. The intuitive remedy for shock\-plus\-complicated\-geometry is adaptive mesh refinement, the standard tool for following moving fronts; here, however, the failure mode is not insufficient resolution but loss of physical realisability\. Standard solvers permit numerical artefacts that correspond to negative pressure or negative density at that single point, at which the simulation crashes\. Choosing the right combination of method, mesh, and time stepping for this regime is the kind of judgement a computational\-fluid\-dynamics specialist makes after consulting several specialized papers and discarding a handful of failed runs\. It is not a textbook lookup\. The system flagged this hazard before any solver invocation, recording in its mesh\-planning report that “under\-resolution there will either quench the shock across too many cells \(smearing\) or crash positivity at the stagnation node” \(Appendix[B\.1](https://arxiv.org/html/2605.11117#A2.SS1)\); the failure mode is identified up front, not discovered after a crash\.

GRAFT\-ATHENA reaches the right configuration without human input\. The top\-NnbrN\_\{\\text\{nbr\}\}neighbors in𝒟\\mathcal\{D\}, weighted by fingerprint similarity and reward, induce a merged action prior over𝒯A\\mathcal\{T\}\_\{A\}\(Fig\.[4](https://arxiv.org/html/2605.11117#S2.F4)B, panels 1 to 3\), and the agent samples through it level by level\. The structural skeleton, namely the governing equations, the boundary\-condition family, and the broad solver class, is already concentrated in the prior, since the neighbors that carry weight all sit in the same compressible\-flow regime as Apollo\. The harsher Mach\-10 regime, however, demands a method that preserves the positivity of pressure and density by construction, and the agent makes that upgrade on its own\. At the chain where this is decided, the proposer reads the per\-chain row together with the node hints attached at build time and the Mach\-10 descriptors ofpp, and the resulting sample lands on a positivity\-preserving method \(Fig\.[4](https://arxiv.org/html/2605.11117#S2.F4)B, rightmost\)\. For this case, the selected production backend was the Trixi\.jl compressible\-flow branch\.

That single choice locks in the rest\. Positivity\-preserving methods are incompatible with adaptive mesh refinement, so the system forgoes adaptivity and compensates with a graded mesh that puts resolution where the shock will be \(Fig\.[4](https://arxiv.org/html/2605.11117#S2.F4)C\); the system explicitly traces this cascade in its own planning record, noting that “IDP is incompatible with AMR … the base mesh must be pre\-graded at plan time and clustered around every feature the physics produces; runtime adaptation is unavailable” \(Appendix[B\.1](https://arxiv.org/html/2605.11117#A2.SS1)\)\. The time\-integration scheme and the numerical fluxes then follow from the same constraint\. To validate the run, the agent itself proposes the diagnostic suite shown in Fig\.[4](https://arxiv.org/html/2605.11117#S2.F4)D: density on a logarithmic scale, local Mach number with theM=1M=1sonic line in red bounding the subsonic stagnation pocket, and a numerical Schlieren of\|∇ρ\|\|\\nabla\\rho\|; this is the same combination a hypersonics specialist would have asked for, screening for shock standoff, stagnation positivity, and shock\-front sharpness in one figure\. The eight individual numerical decisions GRAFT\-ATHENA makes for this case collapse to one physics fact about positivity at the stagnation point, reached by graph traversal rather than retrieval\. This is the form of judgement that ordinarily lives in the experience of numerical analysts and in scattered method papers, reconstructed here autonomously, on a real engineering problem, from a 1968 PDF\. Notably, in the recorded Apollo run the entire chain completed in a single approved iteration: GRAFT\-ATHENA produced this configuration on the first attempt, with no failed run, no advisor\-prescribed correction, and no manual intervention; the diagnostic stage flagged none of the nine candidate failure modes it screens for \(Appendix[B\.1](https://arxiv.org/html/2605.11117#A2.SS1)\)\.

![Refer to caption](https://arxiv.org/html/2605.11117v1/Figures/Capsule.png)Figure 4:Autonomous Mach\-10 solution of the Apollo command module from a 1968 engineering report\.\(A\) Input geometry from the report: dimensioned schematic of the command\-module forebody \(top\) and archival photograph \(bottom\)\. \(B\) From problem to method on the GRAFT substrate: binned fingerprint of the target \(leftmost\), the fingerprint of the closest already\-solved problem underJK⋆J\_\{K^\{\\star\}\}\(Eq\.[11](https://arxiv.org/html/2605.11117#S4.E11)\), SupersonicCylinder \(J=0\.71J=0\.71\), the merged action prior built from the top\-NnbrN\_\{\\text\{nbr\}\}neighbors via the sigmoid\-gated reward blend \(Methods, §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6), Eq\. \([15](https://arxiv.org/html/2605.11117#S4.E15)\)\), and the method GRAFT\-ATHENA selected, traced through𝒯A\\mathcal\{T\}\_\{A\}\(rightmost\)\. \(C\) Agent\-generated mesh \(left\) and self\-audit: neighbor\-growth histogram against the agent’s quality targetg=1\.2g=1\.2\(right\)\. \(D\) Converged solution att≈0\.50,1\.50,3\.00,5\.00t\\approx 0\.50,\\ 1\.50,\\ 3\.00,\\ 5\.00: density on a log scale \(top\), local Mach number with theM=1M=1sonic line in red \(middle\), and numerical Schlierene−K​\|∇ρ\|/\|∇ρ\|maxe^\{\-K\|\\nabla\\rho\|/\|\\nabla\\rho\|\_\{\\max\}\}\(bottom\)\. The middle row shows the Mach number collapsing fromMa∞=10\\mathrm\{Ma\}\_\{\\infty\}=10to a near\-zero subsonic pocket at the stagnation node \(the dark\-blue region behind the shock, bounded by the sonic line\), the regime in which positivity preservation, not adaptivity, is the operative constraint\.
### 2\.6Shear\-thinning rheology of a red\-blood\-cell suspension

We pose the system a clinically grounded particle\-based test: simulate two red\-blood\-cell \(RBC\) suspensions in their natural vessel\-class context and recover the apparent viscosity of each — Gaucher\-disease RBCs at capillary shear, where the slow flow and the disease’s stiffness and aggregation phenotype are most consequential, against healthy RBCs at arteriole shear, the regime in which whole blood is conventionally measuredChaiet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib780),[2026](https://arxiv.org/html/2605.11117#bib.bib775)\]\. The pairing is asymmetric on purpose\. Gaucher disease presents a quantitatively different cell — a stiffer membrane and a raised cell–cell aggregation threshold — and the clinical question is not whether shear thinning can be reproduced in isolation but whether the disease leaves a recognisable rheological fingerprint when each population is placed in the regime that physiologically matters\. The case is a textbook stress test for an agentic framework\. The agent has to commit to a non\-PDE solver family, place a multiscale spectrin\-network membrane on top of it, branch the membrane parameterization into a healthy arm and a disease arm, and choose the wall body force in each arm to match its target vessel class — four interlocking decisions, any one of which collapses the result into the wrong rheological regime if it is off by a step\. The intuitive remedy — dial the cell stiffness or the thermostat until the two viscosities straddle the clinical band — is exactly the failure mode here, since the headline numbers must emerge from the assembled multiscale model rather than be tuned into it\.

The configuration is recovered from the action graph in a single cascade\. From the DPD subtree of𝒢A\\mathcal\{G\}\_\{A\}rooted at LAMMPSThompsonet al\.\[[2022](https://arxiv.org/html/2605.11117#bib.bib63)\], ATHENA selects the multiscale RBC formulation of Pivkin and KarniadakisPivkin and Karniadakis \[[2008](https://arxiv.org/html/2605.11117#bib.bib787)\]and Fedosov and co\-workersFedosovet al\.\[[2010](https://arxiv.org/html/2605.11117#bib.bib788)\], which fixes the membrane substrate as a coarse\-grained spectrin network triangulated on each cell surface\. That single choice locks in the rest\. The substrate forces a worm\-like\-chain bond on every triangulation edge, an area–volume\-conserving angle potential on every triangle, and a bending dihedral on every adjacent triangle pair — selected by the agent as thebreak/ligandbond style and thearea/volumeangle style from the LAMMPS leaves of𝒯A\\mathcal\{T\}\_\{A\}\. A standard DPD pairwise interaction kernel closes the cell–cell and cell–solvent forces, and the assembled simulation cell is a20×20×2020\\\!\\times\\\!20\\\!\\times\\\!20LJ box with frozen wall layers \(atom types 3 and 4\), a periodic outer film \(type 5\) preventing wall\-image mixing, and a core slab populated by deformable RBCs \(type 1\) immersed in a DPD solvent \(type 2\) — the same chip\-flow geometry validated previously for blood biophysics in pathological statesChaiet al\.\[[2023](https://arxiv.org/html/2605.11117#bib.bib778),[2022](https://arxiv.org/html/2605.11117#bib.bib779)\]\. The healthy and Gaucher arms then differ only on the two parameters the disease is known to perturb: the shear elastic modulusEsE\_\{s\}of the spectrin network and the cell–cell disaggregation threshold, with all remaining parameters held to the validated baseline\. The wall body force in each arm is matched to its physiological vessel class —fx=0\.15f\_\{x\}=0\.15at the capillary shear of the Gaucher arm andfx=0\.05f\_\{x\}=0\.05at the arteriole shear of the healthy arm — with a common thermostat target\. The cascade from one biophysical commitment \(multiscale spectrin membrane\) to the eight downstream LAMMPS choices and the two\-knob healthy/Gaucher branch is traced through the action tree rather than retrieved from a configuration template\.

The result is the rheological fingerprint required of any blood\-flow study and, for this pairing, of any in\-silico Gaucher model\. The shear profiles of Fig\.[5](https://arxiv.org/html/2605.11117#S2.F5)A,B are clean enough within the bulk windowz∈\[3,17\]z\\in\[3,17\]for the bulk shear rateγ˙=∂zux\|bulk\\dot\{\\gamma\}=\\partial\_\{z\}u\_\{x\}\|\_\{\\mathrm\{bulk\}\}to be read off as a slope \(−8\.77×10−4\-8\.77\\times 10^\{\-4\}for the Gaucher arm,−1\.36×10−3\-1\.36\\times 10^\{\-3\}for the healthy arm\); the linear\-fit quality is modest in the Gaucher arm \(R2=0\.35R^\{2\}=0\.35\) and comfortable in the healthy arm \(R2=0\.73R^\{2\}=0\.73\), reflecting the larger fluctuation amplitude expected of a stiffer, more aggregated suspension at the lower shear rate where signal\-to\-noise is naturally worse\. We therefore read the Gaucher viscosity as a noisy low\-shear effective estimate, with the direction \(Gaucher above healthy\) and the clinical\-band placement as the robust signals rather than the absolute slope\. The thermostat trace of Fig\.[5](https://arxiv.org/html/2605.11117#S2.F5)C confirms that both runs equilibrate to a constant target temperature within a fraction of a time unit, so neither viscosity reading is contaminated by drift\. After the empirical LJ\-to\-cP conversion that calibrates a single\-Newtonian\-solvent benchmark to physiological plasma viscosity, Fig\.[5](https://arxiv.org/html/2605.11117#S2.F5)D returnsηeff=5\.13\\eta\_\{\\mathrm\{eff\}\}=5\.13cP for the Gaucher capillary arm and3\.313\.31cP for the healthy arteriole arm\. Both values land inside the clinical whole\-blood band of33–1212cP, with the healthy arteriole arm sitting at the band’s lower edge as expected for a less\-loaded, less\-aggregated suspension and the Gaucher capillary arm sitting visibly above it — the two\-arm design should be read as a physiologically matched contrast, not as an isolation of disease parameters at fixed shear: it shows that the assembled GRAFT\-ATHENA DPD model lands in the expected clinical viscosity band and preserves the expected ordering for the healthy\-arteriole and Gaucher\-capillary regimes, recovered from the assembled model and not tuned into it\. The microscopic correlate is in panels F and H: principal\-axis analysis of a single periodic\-unwrapped cell gives an orientation\-invariant aspect ratioλ1/λ2\\lambda\_\{1\}/\\lambda\_\{2\}that captures the cell’s instantaneous deformation independent of tank\-treading orientation, with the high\-shear cell \(λ1/λ2=1\.38\\lambda\_\{1\}/\\lambda\_\{2\}=1\.38\) visibly more elongated than its low\-shear counterpart \(1\.281\.28\) — the cell\-scale companion of the macroscopic viscosity contrast\. We report point estimates rather than block\-bootstrap or replicate\-seed intervals on the bulk slope or propagated viscosity, so the qualitative ordering and clinical\-band placement, not the absolute slope values, are the load\-bearing claims; full uncertainty quantification is deferred to follow\-up\.

The bottom row of Fig\.[5](https://arxiv.org/html/2605.11117#S2.F5)\(panels I–L\) shows the GRAFT machinery that placed this configuration\. The target\-problem fingerprint of the high\-hematocrit Gaucher viscosity case \(panel I\) is matched against𝒟\\mathcal\{D\}, and its closest neighbor is a Poiseuille flow at fingerprint similarityJ=0\.71J=0\.71\(panel J\): a different boundary condition and a different vessel\-class observable, but a fingerprint\-similar problem in the GRAFT geometry, which the policy\-blend of Eq\. \([15](https://arxiv.org/html/2605.11117#S4.E15)\) carries forward\. The merged action prior \(panel K,9797cells with support\) is visibly denser and more diffuse than the Apollo case of §[2\.5](https://arxiv.org/html/2605.11117#S2.SS5), because no single neighbor dominates the blend atJ=0\.71J=0\.71and the prior accumulates support from multiple past entries; the selected method on𝒯A\\mathcal\{T\}\_\{A\}\(panel L,3131active cells\) collapses that prior into a single path through the action tree, the eight LAMMPS choices of the cascade above\. The GRAFT substrate therefore admits a particle\-based fingerprint alongside the PIML and Numerical fingerprints of §[2\.4](https://arxiv.org/html/2605.11117#S2.SS4)and §[2\.5](https://arxiv.org/html/2605.11117#S2.SS5), with the same nearest\-neighbor retrieval, the same policy blend, and the same long\-term\-memory commit\. The DPD branch consequently validates two things at once: that ATHENA carries a non\-PDE solver end to end, with the action tree, the advisor’s diagnostic suite, and the long\-term memory𝒟\\mathcal\{D\}admitting particle\-based fingerprints alongside PIML and Numerical ones; and that the resulting trajectories reproduce the qualitative and quantitative rheology that any blood\-flow study must respect, in line with the broader RBC\-biophysics programme of which this work is a partFedosovet al\.\[[2014](https://arxiv.org/html/2605.11117#bib.bib789)\], Zhanget al\.\[[2020](https://arxiv.org/html/2605.11117#bib.bib776),[2021](https://arxiv.org/html/2605.11117#bib.bib777)\]\.

![Refer to caption](https://arxiv.org/html/2605.11117v1/Figures/DPD_cases.png)Figure 5:ATHENA\-driven DPD shear\-thinning study of a red\-blood\-cell suspension\.All panels share the same simulation geometry, coarse\-graining strategy, and thermostat target; the two main runs differ only in the membrane parameters that the disease perturbs \(shear modulus and cell–cell disaggregation threshold\) and in the wall body force matched to each arm’s physiological vessel class\. The two main runs are reported as the Gaucher capillary case \(red\) and the Healthy arteriole case \(blue\)\. Top row\. \(A\) and \(B\): time\-averaged streamwise velocity⟨ux⟩​\(z\)\\langle u\_\{x\}\\rangle\(z\)for the Gaucher capillary regime \(red\) and the Healthy arteriole regime \(blue\); shaded band marks the bulk fitting windowz∈\[3,17\]z\\\!\\in\\\!\[3,17\]used to extract the shear rate\. \(C\): instantaneous temperature trace; horizontal segments mark the equilibrium\-window mean for each run\. \(D\): apparent viscosityηeff=τx​z/\|∂zux\|bulk\\eta\_\{\\mathrm\{eff\}\}=\\tau\_\{xz\}/\|\\partial\_\{z\}u\_\{x\}\|\_\{\\mathrm\{bulk\}\}in cP after the empirical LJ\-to\-cP calibration; green band is the physiological whole\-blood viscosity range\. Both regimes land inside the clinical band, with the Gaucher capillary case lying above the Healthy arteriole case — the shear\-thinning, hematocrit\-sensitive fingerprint of an RBC suspension\. Middle row \(RBC morphology\)\. Two independent shear\-sweep cases bracketing the operating range: \(E\) fullx​zxz\-projection snapshot atfx=0\.005f\_\{x\}=0\.005\(low shear\), \(F\) zoom of a single periodic\-unwrapped cell at the samefxf\_\{x\}with principal\-axis overlay \(black:λ1\\lambda\_\{1\}, grey:λ2\\lambda\_\{2\}\); \(G\)–\(H\) same pair atfx=0\.5f\_\{x\}=0\.5\(high shear\)\. The reportedλ1/λ2\\lambda\_\{1\}/\\lambda\_\{2\}are orientation\-invariant aspect ratios from a 2\-D principal\-component analysis of the unwrapped point cloud\. Bottom row \(GRAFT fingerprint\+selection quad\)\. \(I\) Target\-problem fingerprint on𝒯P\\mathcal\{T\}\_\{P\}\. \(J\) Closest already\-solved problem in𝒟\\mathcal\{D\}underJK⋆J\_\{K^\{\\star\}\}, Poiseuille atJ=0\.71J=0\.71\. \(K\) Merged action prior on𝒯A\\mathcal\{T\}\_\{A\}built from the top\-NnbrN\_\{\\mathrm\{nbr\}\}neighbors via the sigmoid\-gated reward blend; 97 cells carry non\-zero support\. \(L\) The method GRAFT\-ATHENA selected on𝒯A\\mathcal\{T\}\_\{A\}, 31 active cells\. Panels \(I\)–\(L\) share the0–11grayscale colorbar at right\. The standard DPD pairwise interaction kernel, break/ligand bond style and area/volume angle style — all visible as concentrated mass bands in the merged prior of \(K\) — were inherited from the neighbor and survived the proposer–critic walk to land in \(L\), demonstrating cross\-problem method transfer on a particle\-based, non\-PDE solver\.
### 2\.7From ill\-posed to in\-vivo: autonomous reformulation and solution of a perivascular flow inverse problem\.

![Refer to caption](https://arxiv.org/html/2605.11117v1/Figures/brain.png)Figure 6:Autonomous reformulation and solution of an in\-vivo perivascular\-flow inverse problem on mouse\-brain data\.\(A\) Experimental input from artificial intelligence velocimetry: two\-photon microscopy of the perivascular space \(PVS\) in mouse, with tracer particles tracked from a cisterna\-magna injection under the moving\-boundary formulation; the inverse problem is to recover the full pressure and velocity fields on the PVS geometry from these noisy tracks\. \(B\) From problem to method on the GRAFT substrate: target fingerprint, closest already\-solved problem underJK⋆J\_\{K^\{\\star\}\}atJ=0\.57J=0\.57, merged action prior from the top\-NnbrN\_\{\\text\{nbr\}\}neighbors \(Methods §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6), Eq\. \([15](https://arxiv.org/html/2605.11117#S4.E15)\);164164cells with support\), and the method GRAFT\-ATHENA selected on𝒯A\\mathcal\{T\}\_\{A\}\(6161active cells\); the merged prior is visibly more diffuse than in the Apollo case becauseJK⋆J\_\{K^\{\\star\}\}to the nearest neighbor is low and no individual entry in𝒟\\mathcal\{D\}dominates the blend\. \(C\) Agent\-proposed preprocessing, in nondimensional units including time series ofvobsv\_\{\\text\{obs\}\}\(min\\min,max\\max,mean\\mathrm\{mean\}\) with dashed lines bounding the dominant cycle, and thevobsv\_\{\\text\{obs\}\}scatter at the mid\-plane\. \(D\) Predicted \(red\) versus reference \(blue\) particle tracks at several timesteps\. \(E\) Recovered fields on a cross section of the PVS geometry: pressureppinPa\\mathrm\{Pa\}\(top\) and velocity magnitude∥u∥\\lVert u\\rVertinμ​m/s\\mu\\mathrm\{m\}/\\mathrm\{s\}\(bottom\); magnitudes are consistent with published values\.We consider an inverse problem on in\-vivo data, as defined in the paperBosteret al\.\[[2023](https://arxiv.org/html/2605.11117#bib.bib162)\]\(Fig\.[6](https://arxiv.org/html/2605.11117#S2.F6)A\)\. Tracer particles are injected into the cisterna magna of a mouse and travel through the perivascular space, where two\-photon microscopy records their trajectories on a single observation plane; the moving boundaries of the perivascular geometry are obtained as described inToscanoet al\.\[[2024](https://arxiv.org/html/2605.11117#bib.bib194)\]\. From these single\-plane particle tracks and the available boundary conditions, we aim to reconstruct the velocity and pressure fields on the perivascular geometry\.

The Formalization team \(Methods, §[4\.2\.1](https://arxiv.org/html/2605.11117#S4.SS2.SSS1)\) analyzed the problem\. The proposer returned three simplification candidates, of which the ranker selected the reformulation that enforces conservation of mass as an exact, hard architectural constraint on the recovered velocity field\. The alternatives were discarded on operational grounds\. A pressure\-Poisson / Leray\-projection reduction, for example, was rejected as “operationally inert” in a primitive\-variable PINN, because its scalar residuals are identical to those of the original Stokes formulation and “applying \[the Leray projector\] pointwise requires solving an auxiliary elliptic problem at every collocation point”\. The rederivation under the winning candidate went through cleanly, with continuity collapsing to the vector\-calculus identity∇⋅\(∇×𝐀\)≡0\\nabla\\cdot\(\\nabla\\times\\mathbf\{A\}\)\\equiv 0and the four primitive unknowns\(u,v,w,p\)\(u,v,w,p\)replaced by three potential outputs\(P,R,p\)\(P,R,p\)\.

The well\-posedness audit, however, returned a verdict of ill\-posed and flagged two deficits the new formulation could not absorb on its own\. The first was an additive pressure gaugep↦p\+c​\(t\)p\\mapsto p\+c\(t\), which the agent diagnosed and closed autonomously, inferring the channel axis from the data and prescribing an outlet Dirichlet anchor on the basis that “imposingp​\(t,x,1\.556,z\)=0p\(t,x,1\.556,z\)=0at everyttremoves the additive gaugec​\(t\)c\(t\)entirely”\. The second was an identifiability gap induced by the single\-slab observation geometry, which the agent characterized in terms of an explicit null mode: “any IC mode that has a node at the slab planez∈\[0\.40,0\.42\]z\\in\[0\.40,0\.42\]produces zero observable\(u,v\)\(u,v\)on the slab and is invisible to the data residual”\. Presented with three closure options, the user selected a smoothness regularizer on∂z2\\partial\_\{z\}^\{2\}of the velocity, which suppresses exactly those unobservable modes and entered the formulation as an additional loss term\. The two closures, one autonomous and one user\-elicited, fed forward as inputs to the encode\-select\-solve spine, the audit upgraded the formulation to conditionally well\-posed, and the full interaction trace is recorded in Appendix[B\.2](https://arxiv.org/html/2605.11117#A2.SS2)\.

With the formulation locked in, the encode\-select\-solve spine ran in its standard mode\. The target fingerprint was extracted, the closest already\-solved neighbor was found atJK⋆=0\.57J\_\{K^\{\\star\}\}=0\.57, the merged action prior was assembled over the top\-NnbrN\_\{\\text\{nbr\}\}neighbors, and the method GRAFT\-ATHENA selected on𝒯A\\mathcal\{T\}\_\{A\}is shown in Fig\.[6](https://arxiv.org/html/2605.11117#S2.F6)B; the agent\-side interactions behind this stage are shown in Fig\.[1](https://arxiv.org/html/2605.11117#S1.F1)C and recorded in full in Appendix[B\.2](https://arxiv.org/html/2605.11117#A2.SS2)\. The hard mass\-conservation reformulation and the smoothness regularizer,ℒreg=λreg​𝔼​\[\|∂z2u\|2\+\|∂z2v\|2\+\|∂z2w\|2\]\\mathcal\{L\}\_\{\\rm reg\}=\\lambda\_\{\\rm reg\}\\,\\mathbb\{E\}\\\!\\left\[\\,\|\\partial\_\{z\}^\{2\}u\|^\{2\}\+\|\\partial\_\{z\}^\{2\}v\|^\{2\}\+\|\\partial\_\{z\}^\{2\}w\|^\{2\}\\right\], proposed by the Formalization team is integrated into the action space as a structural input\. Panel C shows the agent\-proposed preprocessing in nondimensional units: thevobsv\_\{\\text\{obs\}\}time series \(max, min, mean\) is used to identify the dominant cycle \(bounded by the dashed lines\), and the lower sub\-panel displays the mid\-plane scatter data\. Panel D shows a quiver\-plot comparison of predicted \(red\) and reference \(blue\) particle tracks across six timesteps, showing close agreement, and panel E plots the recovered fields on the PVS geometry, with pressureppinPa\\mathrm\{Pa\}\(top row\) and velocity magnitude∥u∥\\lVert u\\rVertinμ​m/s\\mu\\mathrm\{m\}/\\mathrm\{s\}\(bottom row\) at three timesteps each\. Notably, the run used a network roughly half the size of the one reported inBosteret al\.\[[2023](https://arxiv.org/html/2605.11117#bib.bib162)\]and converged in only∼105\\sim 10^\{5\}iterations: even though the closest neighbor in𝒟\\mathcal\{D\}was relatively far, the action space pulled a highly specialized stack for this problem, namely a negative\-log\-likelihood \(NLL\) noise\-cleaning networkToscanoet al\.\[[2024](https://arxiv.org/html/2605.11117#bib.bib194)\], vRBA residual weighting and samplingToscanoet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib23)\], adaptive activation functionsJagtapet al\.\[[2020](https://arxiv.org/html/2605.11117#bib.bib141)\], and the SOAP optimizerVyaset al\.\[[2024](https://arxiv.org/html/2605.11117#bib.bib12)\], Wanget al\.\[[2025a](https://arxiv.org/html/2605.11117#bib.bib44)\]\. The recovered velocity attains a relativeL2L^\{2\}track error of approximately30%30\\%against the held\-out reference particles, comparable to the values reported inBosteret al\.\[[2023](https://arxiv.org/html/2605.11117#bib.bib162)\]and within the expected range for an in\-vivo inverse problem on noisy single\-plane data\.

### 2\.8Spectral PINN: an agent\-designed architecture with exponential convergence\.

Beyond producing well\-posed reformulations, the formalization team can exploit the LLM’s vast training knowledge to induce framework or method discovery\. This ability was observed inToscanoet al\.\[[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\], in which the LLM identified several ways to encode exact solutions and propose simplifications\. Building on this feature, we include a simplification agent in the formalization team, in charge of finding exact solutions or ways to simplify the problem\. This agent has access to an analytical knowledge tree embedded into GRAFT\-ATHENA’s internal structure𝒢A\\mathcal\{G\}\_\{A\}\(Fig\.[1](https://arxiv.org/html/2605.11117#S1.F1)B\) and interacts with a ranker agent that scores the proposed candidates \(Methods, §[4\.2\.1](https://arxiv.org/html/2605.11117#S4.SS2.SSS1)\)\. The behavior of this loop, and its dependence on user intervention, is most cleanly illustrated by revisiting the viscous Burgers benchmark of §[2\.4](https://arxiv.org/html/2605.11117#S2.SS4)\.

On this problem the simplification agent returned three candidates from the analytical knowledge tree: an inviscid high\-Reynolds outer limit, a periodic Fourier\-series mode reduction, and a conservative flux\-divergence reformulation\. The inviscid limit was self\-flagged as regime\-incompatible because the characteristic map loses invertibility atts=1/πt\_\{s\}=1/\\piand “cannot represent the viscous internal layer that defines the problem”\. At ranking time the ranker added a fourth candidate of its own, a multi\-harmonic Fourier\-feature input encoding that absorbs periodicity into the architecture at near\-zero implementation cost, and selected it on a “minimal architectural delta from a standard PINN” basis\. This is the formulation already reported as the running Burgers benchmark in §[2\.4](https://arxiv.org/html/2605.11117#S2.SS4)\(Fig\.[3](https://arxiv.org/html/2605.11117#S2.F3)D\)\.

Hidden in the same proposal pass, however, was a second candidate that the rubric had pushed aside: the proposer’s periodic Fourier\-series mode reduction in the spectral\-Galerkin sense, demoted to second place as a high\-implementation\-risk research bet against the cheaper Fourier\-feature route\. On inspection the user queried the agent’s reasoning, and the same ranker articulated what its own scoring had missed against the structure of this specific problem: namely the diagonal viscous damping is “closed\-form, not learned”, removing what is in practice “the hardest part of training a PINN on Burgers, getting the MLP to faithfully representν​ux​x\\nu u\_\{xx\}near the layer”; the IC can also be exactly satisfied, and spatial differentiation reduces to spectral multiplication, removing autodiff inxxentirely\.

The override\-selected reformulation entered the derivation stage\. The proposer expandedu​\(t,x\)=∑n=1N\[an​\(t\)​cos⁡\(n​π​x\)\+bn​\(t\)​sin⁡\(n​π​x\)\]u\(t,x\)=\\sum\_\{n=1\}^\{N\}\[a\_\{n\}\(t\)\\cos\(n\\pi x\)\+b\_\{n\}\(t\)\\sin\(n\\pi x\)\]on a truncation ofNNmodes, which absorbs the periodic boundary condition structurally into the basis, and parameterized the modal coefficients by the hard\-IC encodingan​\(t\)=t​a~n​\(t\)a\_\{n\}\(t\)=t\\tilde\{a\}\_\{n\}\(t\),bn​\(t\)=bnI​C\+t​b~n​\(t\)b\_\{n\}\(t\)=b\_\{n\}^\{IC\}\+t\\tilde\{b\}\_\{n\}\(t\)withb1I​C=−1b\_\{1\}^\{IC\}=\-1, so that the IC enters as a hard constraint rather than a loss term; the nonlinearityu​uxuu\_\{x\}is evaluated pseudospectrally on a dealias grid ofMMpoints and projected back onto theNNretained modes\. The first draft did not survive review: the critic flagged a sign error in the viscous modal term that “would produce anti\-diffusive modal growth rather than viscous damping”, and a second pass tightened the dealiasing rule fromM≥3​NM\\geq 3Nto strictM\>3​NM\>3Non the grounds that “a discrete grid of lengthM=3​NM=3Ncan alias the product mode2​N2Ninto the retained mode−N\-N”\. Finally the loop locked a modal residual for this specific formulation\.

![Refer to caption](https://arxiv.org/html/2605.11117v1/Figures/Spectral_PINN.png)Figure 7:Agent\-designed spectral PINN for viscous Burgers,ν=1/100\\nu=1/100\.\(A\) Architecture locked by the Formalization team: a sine\-only Galerkin truncation with hard IC, diagonal viscous damping, pseudospectral evaluation ofu​uxuu\_\{x\}on a dealias gridM\>3​NM\>3N, and a per\-mode, per\-time vRBA\-weighted MSE as the only active loss\. \(B\) Target problem fingerprint on𝒯P\\mathcal\{T\}\_\{P\}\(5353active cells\), the problem signature against which the action prior is assembled\. \(C\) Action\-tree growth: the initial𝒯A\\mathcal\{T\}\_\{A\}\(156156cells\) and the modified𝒯A\\mathcal\{T\}\_\{A\}\(173173cells\), with1717new leaves \(red\) attached under their parent chains so the tree can host the formalization\-locked decisions; the remainder of the method is drawn from leaves the tree already carried\. \(D\) Dense\-grid predictionu^​\(t,x\)\\hat\{u\}\(t,x\)for the best\-resolved run,N=128N=128\. \(E\) Pointwise field error\|u^−uref\|\|\\hat\{u\}\-u\_\{\\mathrm\{ref\}\}\|\(left\) and per\-mode Galerkin residual\|Rnb​\(t\)\|2\|R\_\{n\}^\{b\}\(t\)\|^\{2\}\(right\) for the same run\. \(F\) Converged per\-mode vRBA tensorλn,i\\lambda\_\{n,i\}on the \(mode, time\) grid\. \(G\) Mode\-count sweep:RL2u\\mathrm\{RL2\}\_\{u\},Lu∞L^\{\\infty\}\_\{u\}, andLu1L^\{1\}\_\{u\}all decay exponentially in the retained\-mode countNN\(‖e‖2∼10−c​N\\\|e\\\|\_\{2\}\\sim 10^\{\-cN\}\), the canonical fingerprint of spectral convergence inherited from the Galerkin basis\.During the derivation the agents flagged the formulation as a “research\-bet” implying that the per\-mode loss sits outside the published PIML toolkit, since standard adaptive reweighting recipes \(RBAAnagnostopouloset al\.\[[2024](https://arxiv.org/html/2605.11117#bib.bib195)\], NTKWanget al\.\[[2022](https://arxiv.org/html/2605.11117#bib.bib472)\]\) are formulated for pointwise spatial residuals and do not transfer mechanically to a mode\-indexed residual axis\. After discussion with the user the team converged on a vector\-RBA \(vRBA\) variant, jointly indexed by modennand time\-collocation pointtit\_\{i\}, on the agent’s reasoning that vRBA “do not care whether the second tensor axis indexes spatial collocation or mode index”\. This per\-mode, per\-time vRBA\-weighted MSE became the only active loss term that the formulation passes into the well\-posedness audit\.

The locked formulation then went into the well\-posedness audit, which derived the standard energy estimate12​dd​t​‖u‖22=−ν​‖ux‖22≤0\\tfrac\{1\}\{2\}\\tfrac\{d\}\{dt\}\\\|u\\\|\_\{2\}^\{2\}=\-\\nu\\\|u\_\{x\}\\\|\_\{2\}^\{2\}\\leq 0and returned a verdict of well\-posed\. In the same pass, however, the audit raised an architectural observation that the previous stage had missed: because the ICu​\(0,x\)=−sin⁡\(π​x\)u\(0,x\)=\-\\sin\(\\pi x\)is parity\-odd and the PDE preserves odd parity under\(x,u\)→\(−x,−u\)\(x,u\)\\to\(\-x,\-u\), uniqueness of the periodic parabolic IBVP forcesu​\(t,−x\)≡−u​\(t,x\)u\(t,\-x\)\\equiv\-u\(t,x\)and hencean​\(t\)≡0a\_\{n\}\(t\)\\equiv 0, so the cosine modes are identically zero and half the network output is wasted\. A critic check verified the closure of the parity\-odd subspace under the dynamics \(u​ux=\(odd\)⋅\(even\)=odduu\_\{x\}=\(\\text\{odd\}\)\\cdot\(\\text\{even\}\)=\\text\{odd\}, so the cosine projections vanish identically\), therefore the live residual count dropped from2​N2NtoNN\. A second observation, a single\-point smoothness gauge att=0\+t=0^\{\+\}, was closed by committing to tanh activations\.

The proposed architecture is described in Fig\.[7](https://arxiv.org/html/2605.11117#S2.F7)A: a sine\-only Galerkin truncation with hard IC, diagonal viscous damping, pseudospectralu​uxuu\_\{x\}on dealias gridM\>3​NM\>3N, and per\-mode, per\-time vRBA\-weighted MSE as the only active loss\. Because the formulation sits well off the standard PIML toolkit, the run was driven interactively by a supporting agent, advancing one stage at a time with the user\. Its first move was to encode the formalization output against the action vocabulary, returning the problem signature of Fig\.[7](https://arxiv.org/html/2605.11117#S2.F7)B and merging the priors over every branch the method touches\. The encoding then hit a wall: several of the formalization’s locked decisions had no leaves in𝒯A\\mathcal\{T\}\_\{A\}to host them, because this method had never been encoded against the action tree\. Rather than downgrade the method, GRAFT\-ATHENA grew the tree, adding1717new nodes \(red, Fig\.[7](https://arxiv.org/html/2605.11117#S2.F7)C\) under their parent chains and registering each as a selectable action for the picker downstream\. Many of the method’s actions, the hard initial\-condition constraint for instance, were already present in𝒯A\\mathcal\{T\}\_\{A\}and only required selection under the standard chains; the1717added leaves are extensions of that existing structure, not a replacement of it\. The method therefore picks from a far larger pool than1717, with the new leaves slotting into the action space alongside the actions the tree already carried\. The vocabulary is not closed: when the formalization outruns it, the existing structure receives the missing leaves and keeps expanding, before any picking, code, or training begins\.

With the action space extended, the encoding team \(see Fig\.[1](https://arxiv.org/html/2605.11117#S1.F1)C\) completed the selection on the modified𝒯A\\mathcal\{T\}\_\{A\}, and the implementation team wrote the associated code, debugged the smoke loop, and ran the production training\. The first iteration returned a relative\-L2L^\{2\}field errorRL2u=1\.107×10−3\\mathrm\{RL2\}\_\{u\}=1\.107\\times 10^\{\-3\}, while the per\-mode PDE loss saturated atLPDE≈8\.3×10−13L\_\{\\mathrm\{PDE\}\}\\approx 8\.3\\times 10^\{\-13\}\. The two numbers look contradictory at first glance, but the diagnostic agent re\-read the gap as a structural feature of the formulation rather than a training pathology: the field\-error figure is “the expected spectral\-truncation gap \(Galerkin loss measures onlyN=48N=48retained modes; field error includes the un\-resolved tail\)”, confirmed by reading\|b48\|2≈10−7\|b\_\{48\}\|^\{2\}\\approx 10^\{\-7\}at the truncation cutoff, energy sitting just outside the retained band\. The flatness signal was reclassifiedFLAT\-on\-floor, “not a training stagnation that more iterations could break, but a truncation symptom that theN→96N\\to 96prescription resolves at the source”\. The advisor then closed the iteration with a single scalar implementation detail, “the remediation is a single hyperparameter bump,N=48→96N=48\\to 96,M=192→320M=192\\to 320, \[…\] no action\-tree edits are warranted: the topology selections all match the user’s non\-negotiable locks, and the diagnostic does not implicate model capacity, optimizer choice, or weighting strategy”\. The full diagnostic and metric trace is recorded in Appendix[B\.4](https://arxiv.org/html/2605.11117#A2.SS4)\.

A second iteration applied the bump and left every other action choice unchanged\. The field error dropped roughly two orders of magnitude toRL2u=6\.598×10−6\\mathrm\{RL2\}\_\{u\}=6\.598\\times 10^\{\-6\}, and the iter\-22advisor’s only further request was a schedule edit \(S2 budget bump\), with no further edits to the action tree\. A two\-orders\-of\-magnitude drop from a single\-knob bump in the modal cutoff is unusual for a PINN, but it is the canonical fingerprint of spectral convergence in traditional numerical methods, which is what the formulation should deliver since the basis is spectral and the discretization is Galerkin\. To explore this behavior, a sweep over the retained\-mode count finds thatRL2u\\mathrm\{RL2\}\_\{u\},Lu∞L^\{\\infty\}\_\{u\}, andLu1L^\{1\}\_\{u\}all decay exponentially inNN\(Fig\.[7](https://arxiv.org/html/2605.11117#S2.F7)G\), confirming spectral convergence\. The best\-resolved run, atN=128N=128, is shown in panels D–F of Fig\.[7](https://arxiv.org/html/2605.11117#S2.F7): dense\-grid prediction \(D\), pointwise field error and per\-mode Galerkin\-residual decomposition \(E\), and the converged per\-mode vRBA tensorλn,i\\lambda\_\{n,i\}\(F\)\. The framework inherits, on this problem, the convergence behavior of the spectral discretization that the formalization team selected, which is the structural payoff of letting the formalization stage own the choice of basis\. Beyond confirming the convergence rate, the curve also indicates that the optimization side is keeping up with the basis at every testedNN: the error decays cleanly with the truncation rather than flattening against an optimizer\-imposed floor\. The same picture was already visible at iter11: atN=48N=48since the PDE loss reached∼10−13\\sim 10^\{\-13\}\. That behavior rests on the action space already carrying a strong second\-order optimizer \(SSBroyden with backtracking line\-search\) and the framework\-specific adaptation of vRBA onto the\(mode,time\)\(\\text\{mode\},\\text\{time\}\)tensor, both selected directly from𝒯A\\mathcal\{T\}\_\{A\}once the picker had access to the1717added leaves\.

## 3Discussion

GRAFT\-ATHENA works by turning method choice into structured memory\. The system is not just a larger language model wrapped around solvers; it is a substrate in which scientific actions have admissible options, dependencies, distances, rewards, and execution traces\. This structure makes the search space inspectable while still allowing the agent to compose methods it has not executed before\.

The representation follows the way scientific computing is already practiced\. A practitioner choosing a numerical method selects attributes that must hold together, commits to one option inside each attribute, and respects cross\-rules between them: a flux can constrain a limiter, a representation can constrain a loss, and an inverse problem can constrain the regularization it can support\. GRAFT\-ATHENA makes these relations explicit instead of leaving them in prose, prompts, or code comments\. Because the right method is uncertain before execution, the policy must remain probabilistic; exploration is the mechanism by which useful combinations are found\.

The probabilistic formulation would be unusable if every decision were sampled jointly\. Conditioning on a problem couples the choices in the action graph, but only some of those choices are coupled in practice\. GRAFT exploits that asymmetry: independent chains are sampled locally, while documented cross\-chain rules remain explicit inℛ\\mathcal\{R\}\. The resulting policy footprint scales with the chain\-local choices plus the stored cross\-rules, rather than with the full product over all choices\. The factorization is therefore a consequence of the dependency structure, not an arbitrary compression\.

The finite\-tree representation should be read in the same operational sense\. It is not a claim that scientific invention is finite, or that every possible architecture and parameterization can be listed in advance\. It is a claim about the solver\-facing choices available to an autonomous agent at a given time: these choices are organized into a finite set of families, ranges, constraints, and admissible combinations\. When the available practice changes, the tree can change with it\. The spectral PINN example expands the tree during a derivation, while the Trixi\.jl and Nektar\+\+ examples expand it from documentation\.

GRAFT is related to retrieval\-augmented generation and knowledge\-graph approaches, but it assigns a different role to stored information\. Retrieval supplies context, and knowledge graphs organize entities and relations; GRAFT turns information into executable structure\. The same documentation that can be retrieved as text becomes admissible actions, explicit dependencies, metric neighborhoods, scoreable rewards, and update rules\. Proposition[4\.1](https://arxiv.org/html/2605.11117#S4.Thmtheorem1)is important for this reason: it connects the substrate to classical factorized probabilistic representationsPearl \[[1988](https://arxiv.org/html/2605.11117#bib.bib781)\]and shows that the factored representation preserves the documented couplings inℛ\\mathcal\{R\}, so compression does not silently erase the constraints that make a method coherent\.

The ingestion experiments work because production solvers already contain this structure in their documentation\. Trixi\.jl and Nektar\+\+ do not merely list commands; they expose option families, admissibility constraints, solver interactions, and practical expert guidance near the relevant choices\. GRAFT\-ATHENA converts that material into nodes, cross\-rules, and node\-attached hints that can be traversed by the proposer and checked by the critic\. The Trixi and Nektar\+\+ ingestions, with 51 and 60 cross\-rules respectively, show that the substrate can absorb a real solver vocabulary without hand\-coding a bespoke agent for each solver\.

The comparison with the earlier agentic systemToscanoet al\.\[[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\]shows what changes when expert knowledge moves from a prompt into a persistent action substrate\. This studies showed that expert methodological priors can make agentic scientific computing effective, but its scaffolding lived largely in prompts and examples available to the model\. GRAFT\-ATHENA stores that scaffolding as traversable choices with dependencies, priors, rewards, and outcomes\. The canonical PIML benchmarks therefore test more than raw performance: they test whether structured method memory improves familiar cases where the earlier agent was already competent\. The Burgers Reynolds\-continuation case is especially informative because the substrate retrieved and reused a continuation strategy even though the target regime was only moderately stiff\.

The Apollo calculation illustrates the same mechanism in a harder setting\. The nearest prior case had a similarity of about0\.710\.71, enough to orient the search but not enough to determine the final method\. The decisive ingredient was the combination of that prior with rich Trixi\.jl documentation, node\-level solver hints, and a nine\-level decision tree produced during the run\. Positivity preservation became a central Mach\-10 constraint, but it was handled inside a broader set of coupled choices involving discretization, limiting, meshing, time stepping, and solver admissibility\. The one\-shot result is therefore a constrained adaptation guided by metric memory and solver documentation\.

This division of labor also explains why the LLM remains necessary\. The closed\-form prior ranks regions of the action space and transfers information from nearby solved problems, but it does not by itself read documentation, interpret node hints, or assemble a coherent assignment under multiple constraints\. The LLM proposer performs that local synthesis, and the critic evaluates the proposal against the same substrate\. In this architecture, the LLM is not a replacement for the metric prior or the action graph; it is the mechanism that makes the structured prior usable in large, partially documented method spaces\.

The broad empirical performance follows from the same division of labor\. GRAFT\-ATHENA does well because the action space contains strong method primitives and makes their combinations reusable: SSBroyden with backtracking, vRBA, spectral bases, hard initial\-condition encodings, NLL noise cleaning, SOAP, and adaptive activations are not isolated methods but selectable components with context\. The LLM composes these components under the constraints of𝒯A\\mathcal\{T\}\_\{A\}, while the metric prior biases the composition toward combinations that have worked on nearby problems\. This explains why the same system lowers error floors on canonical PIML benchmarks, solves the Apollo configuration in one run, converges on the perivascular inverse problem with a smaller network and∼105\\sim 10^\{5\}iterations, and supports exponential convergence in the spectral PINN case\. The common pattern is reusable method structure coupled to execution feedback\.

The problem representation is as important as the method representation\. An agent can spend substantial compute searching over methods for a problem that is ill posed, under\-specified, or mismatched to the available action vocabulary\. The Formalization team sits upstream of the sampler to prevent this failure mode: it audits the problem graph before the method search begins and proposes reformulations when the original statement cannot support stable inference\. In the perivascular case, the autonomous gauge fix and user\-elicited smoothness regularizer converted an ill\-posed inverse problem into a conditionally well\-posed one\. That change did not merely improve a selected method; it changed the search problem the agent was solving\.

The spectral PINN case illustrates a concrete mode of method discovery: synthesis of known mathematical primitives into a new trainable solver\. This is how much of numerical\-method research proceeds in practice\. Related neural–spectral strategies have also been proposed by human researchersRaynaudet al\.\[[2022](https://arxiv.org/html/2605.11117#bib.bib8)\], Yuet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib6)\], Duet al\.\[[2023](https://arxiv.org/html/2605.11117#bib.bib7)\], Meuriset al\.\[[2023](https://arxiv.org/html/2605.11117#bib.bib4)\], and classical spectral methods already establish the convergence behavior such representations can inheritBasdevantet al\.\[[1986](https://arxiv.org/html/2605.11117#bib.bib5)\]\. The discovery here is not the isolated observation that Fourier or modal coordinates are useful, but the construction of a sine\-only Galerkin PINN with hard initial data, diagonal viscous damping, dealiased modal nonlinearities, and per\-mode vRBA as the active residual objective\. That combination was not encoded in the prior PIML action space as a selectable method before the run, and it produced the qualitative signature expected of a genuinely spectral construction, namely exponential convergence with retained mode count\. Once validated, the method is no longer an isolated discovery event; it becomes part of the action memory available to future problems\.

The same structure that makes such accumulation possible also makes the limitations clear\. The prior degrades when𝒟\\mathcal\{D\}is sparse and no close neighbor exists; ingestion quality depends on the richness and accuracy of the target solver’s documentation; each new entry in𝒟\\mathcal\{D\}requires a full method evaluation that can cost hours to days; and the proposer and critic still rely on the local judgment of an LLM\. These are not incidental implementation issues\. They define the next engineering and scientific requirements: denser memory in cold domains, stronger independent critics, better documentation audits, and more complete traces of how proposals were generated and rejected\.

Those traces matter because the present system does not yet identify full Pearl Level\-3 counterfactuals\. The logs do, however, record pre\-execution diagnostic reasoning that points in that direction\. In the Apollo run, the planner anticipated that insufficient resolution would threaten positivity near the stagnation region before the solver was called, which shaped the mesh and stabilization choices\. In the spectral PINN derivation, the ranker recognized that vRBA could be reused when the second tensor axis indexed modes rather than spatial collocation points\. These are diagnostic sensitivity judgments, not identified counterfactuals; supporting the latter will require richer traces of proposal state, seeds, runtime environment, rejected branches, and execution outcomes\.

GRAFT\-ATHENA therefore does not make autonomous scientific computing powerful by relying only on a larger language model\. It makes the search space inspectable, metric, and cumulative\. In the present work that structure supports association and intervention: nearby problems bias priors, selected methods are executed, and outcomes return to memory\. The next step is richer trace logging, so that the same geometry can support genuine counterfactual questions about methods not taken\. In that sense, GRAFT\-ATHENA is a first implementation of geometric memory for autonomous scientific computing, with a substrate that plausibly transfers to method spaces well outside continuum physicsZahavy \[[2026](https://arxiv.org/html/2605.11117#bib.bib10)\], Braga\-Neto \[[2026](https://arxiv.org/html/2605.11117#bib.bib802)\]\.

## 4Methods

### 4\.1GRAFT: Graph Reduction to Adaptive Factored Trees

#### 4\.1\.1Setup

We model ATHENA as a stochastic policy

π:𝒫→Δ​\(ℳ\),\\pi:\\mathcal\{P\}\\to\\Delta\(\\mathcal\{M\}\),where𝒫\\mathcal\{P\}is the space of admissible problems \(PDE specifications, inverse\-data problems, and particle\-dynamics cases in our setting\),ℳ\\mathcal\{M\}is the space of admissible methods, andΔ​\(ℳ\)\\Delta\(\\mathcal\{M\}\)is the probability simplex overℳ\\mathcal\{M\}\. Concretely,π​\(m∣p\)\\pi\(m\\mid p\)is the probability of selecting methodmmon problempp\. When deployment requires a single answer, we take the deterministic readout

m^​\(p\)=arg⁡maxm∈ℳ⁡π​\(m∣p\),\\hat\{m\}\(p\)=\\arg\\max\_\{m\\in\\mathcal\{M\}\}\\pi\(m\\mid p\),i\.e\., the mode ofπ\(⋅∣p\)\\pi\(\\cdot\\mid p\)\.

The method spaceℳ\\mathcal\{M\}is built from an action vocabulary𝒜\\mathcal\{A\}, structured as a directed knowledge graph𝒢A=\(V,EA\)\\mathcal\{G\}\_\{A\}=\(V,E\_\{A\}\)with vertex setV=𝒜V=\\mathcal\{A\}\. Each vertex is an individual action, for instance “use Adam,” “use KAN,” or “use Fourier features\.” Notice that, in this sense, the action tree decomposes naturally intokkchains, each a subtree rooted at anss\-decision \(a “pick one” choice\) with leaf alphabet𝒜j⊆𝒜\\mathcal\{A\}\_\{j\}\\subseteq\\mathcal\{A\}, the set of action choices available at chainjj\. A method is then a tuple

m=\(a1,…,ak\),aj∈𝒜j∪\{∅j\},m=\(a\_\{1\},\\dots,a\_\{k\}\),\\qquad a\_\{j\}\\in\\mathcal\{A\}\_\{j\}\\cup\\\{\\varnothing\_\{j\}\\\},where the null action∅j\\varnothing\_\{j\}marks chains left inactive because an upstream choice did not open them\. The full method space

ℳ⊆∏j=1k\(𝒜j∪\{∅j\}\)\\mathcal\{M\}\\;\\subseteq\\;\\prod\_\{j=1\}^\{k\}\\bigl\(\\mathcal\{A\}\_\{j\}\\cup\\\{\\varnothing\_\{j\}\\\}\\bigr\)is the subset of tuples consistent with the dependency and structural\-nesting rules introduced below\.

The policyπ\\piis informed by past runs, recorded as the observation set𝒟=\{\(pi,mi,Oi,ri\)\}i=1N\\mathcal\{D\}=\\\{\(p\_\{i\},m\_\{i\},O\_\{i\},r\_\{i\}\)\\\}\_\{i=1\}^\{N\}, wherepi∈𝒫p\_\{i\}\\in\\mathcal\{P\}is the problem,mi=\(ai,1,…,ai,k\)∈ℳm\_\{i\}=\(a\_\{i,1\},\\dots,a\_\{i,k\}\)\\in\\mathcal\{M\}the method used to solve it, andOiO\_\{i\}a tuple of method\-family\-specific observables recorded during the run\. For instance for PIML a potential choice isOi=\(ti,ei\)O\_\{i\}=\(t\_\{i\},e\_\{i\}\)withtit\_\{i\}the wall\-clock time andeie\_\{i\}the relativeL2L^\{2\}error; for numerical solvers,OiO\_\{i\}may bundle conditioning, entropy, iteration count, and residual; for data\-driven methods, throughput and prediction error\. The observables carry the information needed to qualifymim\_\{i\}through a quantitative metric, the rewardri∈\[0,rmax\]r\_\{i\}\\in\[0,r\_\{\\max\}\], withrmax=100r\_\{\\max\}=100fixed by construction \(§[4\.1\.7](https://arxiv.org/html/2605.11117#S4.SS1.SSS7)\)\. Notice thatrir\_\{i\}ranks methods under the chosen reward metric: high\-reward entries in𝒟\\mathcal\{D\}reinforce successful\(p,m\)\(p,m\)pairs, while low\-reward entries register weak performance and contribute little positive reinforcement to nearby aggregations through the non\-negative weighting of §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)\.

The reward rubric inherits the four\-axis structure of ATHENAToscanoet al\.\[[2025a](https://arxiv.org/html/2605.11117#bib.bib18)\]\(accuracy, efficiency, details, optimality\) and adapts the scoring to an expected\-value setting: rather than ranking the run against the empirical maximum observed across past runs, each axis is scored against the expected value calibrated from the nearest neighbors ofppin𝒟\\mathcal\{D\}\(§[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)\), so the reward responds to performance relative to what comparable problems in the corpus have achieved\. The accuracy axis and the flatness component of the details axis are computed analytically fromOiO\_\{i\}\(the relativeL2L^\{2\}error against the reference, the residual flatness on the converged trace\); the remaining components are scored by the Advisor team \(§[4\.2](https://arxiv.org/html/2605.11117#S4.SS2)\) under per\-family rubrics, with no live human\-in\-the\-loop step at runtime\. The exact per\-family weights and analytic forms vary by problem class and are tracked in the run logs alongside𝒟\\mathcal\{D\}\.

#### 4\.1\.2Graph reduction: from DAG to tree

Although the formulation above is consistent, working with𝒢A\\mathcal\{G\}\_\{A\}directly is impractical\. A graph is convenient as a picture but cumbersome to operate on: in a DAG, a node reachable through several parents inherits a different role on each incoming path, so the information attached to it depends on context rather than on the node itself\. To recover an independent meaning for each node, we project𝒢A\\mathcal\{G\}\_\{A\}onto a tree𝒯\\mathcal\{T\}, on which every node has a single parent, every method corresponds to one root\-to\-leaf path per chain, and each node lands at a unique location under the embedding of §[4\.1\.5](https://arxiv.org/html/2605.11117#S4.SS1.SSS5)\. The cross\-edge information sacrificed in the projection is preserved as a separate set of rules, introduced below\.

The same construction applies to the problem graph𝒢P\\mathcal\{G\}\_\{P\}, so we describe it generically for an arbitrary knowledge graph𝒢\\mathcal\{G\}and instantiate it independently on𝒢P\\mathcal\{G\}\_\{P\}and𝒢A\\mathcal\{G\}\_\{A\}to obtain𝒯P\\mathcal\{T\}\_\{P\}and𝒯A\\mathcal\{T\}\_\{A\}\. Two edge types in𝒢\\mathcal\{G\}are read directly from the underlying decisions:characterized\_by, denotedccand meaning “all of these apply”, is the multi\-attribute family\-membership backbone, andsubdivides\_in, denotedssand meaning “pick one”, is the mutually exclusive discriminative decision that constitutes a chain alphabet𝒜j\\mathcal\{A\}\_\{j\}\. The reduction selects a spanning tree of𝒢\\mathcal\{G\}rooted at its root, so that every non\-root vertex retains exactly one parent and multi\-parent links are broken; for instance, a Fourier\-features node reachable from both “periodic domain” and “spectral architecture” keeps a single canonical parent\. The resulting tree

𝒯=\(V𝒯,E𝒯\)\\mathcal\{T\}=\(V\_\{\\mathcal\{T\}\},\\ E\_\{\\mathcal\{T\}\}\)has its internal vertices given by thecc\- andss\-decision nodes, i\.e\., the chain alphabets\{𝒜j\}j=1k\\\{\\mathcal\{A\}\_\{j\}\\\}\_\{j=1\}^\{k\}together with thecc\-categories that aggregate them, and its leaf vertices given by the actionsa∈𝒜ja\\in\\mathcal\{A\}\_\{j\}\. Consequently, a methodm=\(a1,…,ak\)m=\(a\_\{1\},\\dots,a\_\{k\}\)withaj∈𝒜ja\_\{j\}\\in\\mathcal\{A\}\_\{j\}is precisely a leaf\-per\-chain selection on𝒯\\mathcal\{T\}\.

#### 4\.1\.3Encoding dependencies

The tree𝒯\\mathcal\{T\}exposes a natural unit of decision below the chain alphabets themselves\. Cutting𝒯\\mathcal\{T\}at everycc\-node other than its own root partitions the tree into a forest whose pieces are rooted atcc\-nodes, contain onlyss\-decision interior vertices, and terminate at leaves or at the nextcc\-node down\. We call each such piece a subchain and write𝒞=\{C1,…,C\|𝒞\|\}\\mathcal\{C\}=\\\{C\_\{1\},\\dots,C\_\{\|\\mathcal\{C\}\|\}\\\}for the resulting set\. A subchain is a small Markov decision process: at eachss\-node along it the policy picks one child, the picked child becomes the nextss\-node, and the process terminates when a leaf or anothercc\-node is reached\. Subchains are the natural unit on whichπ\\piacts, and the chain alphabets\{𝒜j\}j=1k\\\{\\mathcal\{A\}\_\{j\}\\\}\_\{j=1\}^\{k\}are recovered as the leaf alphabets of the subchains that terminate in leaves\. Each tree node lies in exactly one subchain, recorded by the enclosing\-chain mapν:V​\(𝒯\)→𝒞∪\{⊥\}\\nu:V\(\\mathcal\{T\}\)\\to\\mathcal\{C\}\\cup\\\{\\bot\\\}, where⊥\\botis a sentinel marking the absence of an enclosing chain \(ν​\(u\)=⊥\\nu\(u\)=\\botonly at the global root\)\. The nesting of subchains within𝒯\\mathcal\{T\}is recorded by the nearest\-enclosingcc\-parent mapρ:𝒞→𝒞∪\{⊥\}\\rho:\\mathcal\{C\}\\to\\mathcal\{C\}\\cup\\\{\\bot\\\}: for each subchainCCrooted below acc\-node,ρ​\(C\)\\rho\(C\)is the first chain re\-entered by walking up alongcc\-edges fromCC’s root, or⊥\\botif the global root is reached without crossing into another chain\. The pair\(ν,ρ\)\(\\nu,\\rho\)records the structural nesting of𝒯\\mathcal\{T\}\.

The cross\-edges discarded by the spanning\-tree projection of the previous section are still needed: they carry the couplings between distinct subchains that𝒯\\mathcal\{T\}alone cannot represent\. We collect them as a rule setℛ\\mathcal\{R\}and defer the formal structure of an individual rule to the operator definitions below; for nowℛ\\mathcal\{R\}is the explicit list of cross\-subchain couplings between the subchains in𝒞\\mathcal\{C\}\.

A direct way to use this structure would be to flatten𝒯\\mathcal\{T\}into a single decision step over the full leaf tuplem=\(a1,…,ak\)∈𝒜1×⋯×𝒜km=\(a\_\{1\},\\dots,a\_\{k\}\)\\in\\mathcal\{A\}\_\{1\}\\times\\cdots\\times\\mathcal\{A\}\_\{k\}, modelingπ\\pias a distribution on the Cartesian product\. The formulation is correct but treats every leaf tuple as its own atomic action, so the action space grows as∏j\|𝒜j\|\\prod\_\{j\}\|\\mathcal\{A\}\_\{j\}\|, exponentially in the number of subchains, and the rules inℛ\\mathcal\{R\}are enforced only by carving forbidden tuples out of the product after the fact rather than by structuring the decision\.

Observe instead that acc\-node carries the meaning “all of these apply”, so its descendant subchains are co\-instantiated as parallel attributes rather than alternatives\. We model them as conditionally independent in the absence of rules, and the rules inℛ\\mathcal\{R\}collect exactly the cross\-chain couplings the model chooses not to drop\. Most pairs of subchains are therefore taken to be independent, and the few that are not are linked by an explicit cross\-edge whose endpoints we know\. The flat product wastes this structure by treating every pair as if it were coupled\.

The easiest way to encode this observation would be to declare the subchains outright independent and write the policy asπ​\(m\)=∏j=1kπj​\(aj\)\\pi\(m\)=\\prod\_\{j=1\}^\{k\}\\pi\_\{j\}\(a\_\{j\}\), with eachπj∈Δ​\(𝒜j\)\\pi\_\{j\}\\in\\Delta\(\\mathcal\{A\}\_\{j\}\)taken from𝒯\\mathcal\{T\}\. Notice that under this construction the action space collapses from\|ℳ0\|\|\\mathcal\{M\}\_\{0\}\|to a sum ofkksmall distributions, but the cross\-chain couplings collected inℛ\\mathcal\{R\}disappear with it, and those couplings are precisely what records that a choice on one chain can change what is admissible on another, which we cannot afford to lose\. However, most of the structural information needed to assemble a working method lives inℛ\\mathcal\{R\}, so the goal is a factorization that keeps those couplings while still reducing the action space below the naive Cartesian product\.

#### 4\.1\.4The action space and its factorization

Notice that a useful factorization must reduce the joint while simultaneously preserving the constraints inℛ\\mathcal\{R\}at the chains where they fire\. We achieve this by lifting each rule into a local edit on the affected chain’s distribution\. In this setting, a ruleRℓ∈ℛR\_\{\\ell\}\\in\\mathcal\{R\}is a four\-tuple

Rℓ=\(hℓ,Tℓtrig,Tℓtgt,eℓ\),R\_\{\\ell\}\\;=\\;\\bigl\(h\_\{\\ell\},\\ T^\{\\mathrm\{trig\}\}\_\{\\ell\},\\ T^\{\\mathrm\{tgt\}\}\_\{\\ell\},\\ e\_\{\\ell\}\\bigr\),\(1\)wherehℓh\_\{\\ell\}is the natural\-language sentence from which the rule is lifted,TℓtrigT^\{\\mathrm\{trig\}\}\_\{\\ell\}is a tuple of values on chains other than the affected one whose presence activates the rule,Tℓtgt⊆𝒜j​\(ℓ\)T^\{\\mathrm\{tgt\}\}\_\{\\ell\}\\subseteq\\mathcal\{A\}\_\{j\(\\ell\)\}is the target slice on the affected chainj​\(ℓ\)j\(\\ell\), andeℓ∈\{ℱ,𝒵\}e\_\{\\ell\}\\in\\\{\\mathcal\{F\},\\mathcal\{Z\}\\\}is the effect, namely force or zero out\. For instance, in Trixi\.jl the documented constraint that the Galerkin formulationMHD\-GLMrequiressurface\_fluxto be one of \{LLF,HLLE\} lifts to a rule withTtrig=\{MHD\-GLM\}T^\{\\mathrm\{trig\}\}=\\\{\\texttt\{MHD\-GLM\}\\\}on the equation chain,Ttgt=\{LLF,HLLE\}T^\{\\mathrm\{tgt\}\}=\\\{\\texttt\{LLF\},\\texttt\{HLLE\}\\\}on the surface\_flux chain, and effectℱ\\mathcal\{F\}\. Each rule contributes, for everyt∈Tℓtrigt\\in T^\{\\mathrm\{trig\}\}\_\{\\ell\}, a directed edgeν​\(t\)→j​\(ℓ\)\\nu\(t\)\\to j\(\\ell\)between distinct chains, and we collect these into the rule\-induced edge setEEon𝒞\\mathcal\{C\}\(dropping anyν​\(t\)=j​\(ℓ\)\\nu\(t\)=j\(\\ell\)self\-loop, which contributes nothing\)\.

Notice that the two effects𝒵\\mathcal\{Z\}andℱ\\mathcal\{F\}correspond to the way a domain expert is most likely to phrase each constraint:𝒵\\mathcal\{Z\}for “this combination is not allowed” andℱ\\mathcal\{F\}for “this combination requires that”\. Formally, given a current distributionπj∈Δ​\(𝒜j\)\\pi\_\{j\}\\in\\Delta\(\\mathcal\{A\}\_\{j\}\)on the leaves of chainjjand a target sliceT⊆𝒜jT\\subseteq\\mathcal\{A\}\_\{j\},

𝒵​\(πj;T\)​\(v\)=\{0v∈T,πj​\(v\)1−∑w∈Tπj​\(w\)v∉T,\\mathcal\{Z\}\(\\pi\_\{j\};\\,T\)\(v\)\\;=\\;\\begin\{cases\}0&v\\in T,\\\\\[2\.0pt\] \\dfrac\{\\pi\_\{j\}\(v\)\}\{1\-\\sum\_\{w\\in T\}\\pi\_\{j\}\(w\)\}&v\\notin T,\\end\{cases\}\(2\)ℱ​\(πj;T\)​\(v\)=\{πj​\(v\)∑w∈Tπj​\(w\)v∈T,0v∉T,\\mathcal\{F\}\(\\pi\_\{j\};\\,T\)\(v\)\\;=\\;\\begin\{cases\}\\dfrac\{\\pi\_\{j\}\(v\)\}\{\\sum\_\{w\\in T\}\\pi\_\{j\}\(w\)\}&v\\in T,\\\\\[2\.0pt\] 0&v\\notin T,\\end\{cases\}\(3\)where𝒵\\mathcal\{Z\}zeroes the target mass and renormalises over the survivors andℱ\\mathcal\{F\}does the dual, restricting support to the target and renormalising there\. The two are duals under set complement,ℱ​\(πj;T\)=𝒵​\(πj;𝒜j∖T\)\\mathcal\{F\}\(\\pi\_\{j\};\\,T\)=\\mathcal\{Z\}\(\\pi\_\{j\};\\,\\mathcal\{A\}\_\{j\}\\setminus T\)\. Both operators are well defined whenever the surviving target carries positive pre\-mass:𝒵​\(πj;T\)\\mathcal\{Z\}\(\\pi\_\{j\};\\,T\)requiresπj​\(𝒜j∖T\)\>0\\pi\_\{j\}\(\\mathcal\{A\}\_\{j\}\\setminus T\)\>0, andℱ​\(πj;T\)\\mathcal\{F\}\(\\pi\_\{j\};\\,T\)requiresπj​\(T\)\>0\\pi\_\{j\}\(T\)\>0\. Sequential composition is order\-independent on the open set where every intermediate denominator is positive; if cumulative effects exhaust support, the rule set is internally inconsistent at that trigger and the build aborts\.

Notice that the effect of these operators can be combined\. By construction, three properties ofℱ\\mathcal\{F\}and𝒵\\mathcal\{Z\}make them safe to apply in sequence on the same chain\. \(1\) A leaf that survives keeps post\-mass proportional to its pre\-mass, so the policy’s preference ordering inside the allowed set is untouched\. \(2\) Both operators act on𝒜j\\mathcal\{A\}\_\{j\}alone \(the trigger encodes where the constraint comes from, the operator only enforces it locally\), so the per\-chain factorisation is preserved\. \(3\) Multiple rules firing on the same chain commute under composition: two zero\-outs reduce to support intersection, two forces likewise, and a force followed by a zero\-out lands onTFtgt∖TZtgtT^\{\\mathrm\{tgt\}\}\_\{F\}\\setminus T^\{\\mathrm\{tgt\}\}\_\{Z\}with the original mass ratios irrespective of order\. The composite distribution at sample time is therefore well defined regardless of application order, on the open set where the per\-step support condition above holds\. If the cumulative effect leaves empty support, the build aborts with the offending pair as a witness\.

With operators in hand, the chains can be organised so that every rule’s triggers are resolved before the chain it acts on\. Collecting the rule\-induced edgesEEtogether with the structural\-nesting edgesEρ=\{ρ​\(C\)→C:C∈𝒞,ρ​\(C\)≠⊥\}E\_\{\\rho\}=\\\{\\rho\(C\)\\to C:C\\in\\mathcal\{C\},\\ \\rho\(C\)\\neq\\bot\\\}from §[4\.1\.3](https://arxiv.org/html/2605.11117#S4.SS1.SSS3)gives the chain dependency graphH=\(𝒞,E∪Eρ\)H=\(\\mathcal\{C\},\\,E\\cup E\_\{\\rho\}\), and the level of a chain is its longest\-path length inHH, well\-defined wheneverHHis acyclic \(checked at build time, below\)\. Equivalently, a level is a maximal set of chains whose triggers all reference values on chains assigned to strictly lower levels\.

Under this construction, sampling proceeds level by level: at each chainjjactive at the current level, the priorπj\\pi\_\{j\}is taken from𝒯\\mathcal\{T\}, every ruleRℓR\_\{\\ell\}withj​\(ℓ\)=jj\(\\ell\)=jwhose trigger has been resolved by previously sampled values is applied through its effect operator \(in any order, by commutativity\), andaja\_\{j\}is sampled from the resulting distribution

π~j\(⋅\|a<level​\(j\)\)=eℓr\(⋯eℓ1\(πj;Tℓ1tgt\)⋯;Tℓrtgt\),\\widetilde\{\\pi\}\_\{j\}\\bigl\(\\,\\cdot\\,\\bigm\|\\,a\_\{<\\mathrm\{level\}\(j\)\}\\bigr\)\\;=\\;e\_\{\\ell\_\{r\}\}\\\!\\Bigl\(\\,\\cdots\\,e\_\{\\ell\_\{1\}\}\\bigl\(\\pi\_\{j\};\\,T^\{\\mathrm\{tgt\}\}\_\{\\ell\_\{1\}\}\\bigr\)\\cdots\\,;\\,T^\{\\mathrm\{tgt\}\}\_\{\\ell\_\{r\}\}\\Bigr\),\(4\)where\{ℓ1,…,ℓr\}\\\{\\ell\_\{1\},\\dots,\\ell\_\{r\}\\\}are the rules whose triggers are met by the lower\-level valuesa<level​\(j\)a\_\{<\\mathrm\{level\}\(j\)\}\. To assign mass to the inactive case, writeactj​\(a<level​\(j\)\)∈\{0,1\}\\mathrm\{act\}\_\{j\}\(a\_\{<\\mathrm\{level\}\(j\)\}\)\\in\\\{0,1\\\}for the indicator that chainjjis opened by the lower\-level picks \(determined by the structural nesting of𝒯\\mathcal\{T\}and anyℱ/𝒵\\mathcal\{F\}/\\mathcal\{Z\}effects inℛ\\mathcal\{R\}that closejjoutright\), and extend the per\-chain kernel to the augmented alphabet𝒜j∪\{∅j\}\\mathcal\{A\}\_\{j\}\\cup\\\{\\varnothing\_\{j\}\\\}by

π¯j​\(aj\|a<level​\(j\)\)=\{π~j​\(aj\|a<level​\(j\)\),actj=1​and​aj∈𝒜j,1,actj=0​and​aj=∅j,0,otherwise\.\\bar\{\\pi\}\_\{j\}\\bigl\(a\_\{j\}\\bigm\|a\_\{<\\mathrm\{level\}\(j\)\}\\bigr\)\\;=\\;\\begin\{cases\}\\widetilde\{\\pi\}\_\{j\}\\bigl\(a\_\{j\}\\bigm\|a\_\{<\\mathrm\{level\}\(j\)\}\\bigr\),&\\mathrm\{act\}\_\{j\}=1\\text\{ and \}a\_\{j\}\\in\\mathcal\{A\}\_\{j\},\\\\\[2\.0pt\] 1,&\\mathrm\{act\}\_\{j\}=0\\text\{ and \}a\_\{j\}=\\varnothing\_\{j\},\\\\\[2\.0pt\] 0,&\\text\{otherwise\.\}\\end\{cases\}\(5\)The full policy factorises, for fixed problemppand fixed repository𝒟\\mathcal\{D\}, as

π​\(m∣p,𝒟\)=∏j=1kπ¯j​\(aj\|a<level​\(j\),p,𝒟\),\\pi\(m\\mid p,\\mathcal\{D\}\)\\;=\\;\\prod\_\{j=1\}^\{k\}\\bar\{\\pi\}\_\{j\}\\bigl\(a\_\{j\}\\bigm\|a\_\{<\\mathrm\{level\}\(j\)\},\\,p,\\,\\mathcal\{D\}\\bigr\),\(6\)where each factor is the extended per\-chain kernel of Eq\. \([5](https://arxiv.org/html/2605.11117#S4.E5)\), withπ~j\\widetilde\{\\pi\}\_\{j\}from Eq\. \([4](https://arxiv.org/html/2605.11117#S4.E4)\) on its active branch\. The construction reduces the action space from\|ℳ0\|\|\\mathcal\{M\}\_\{0\}\|to a sum ofkksmall distributions while preserving the cross\-chain constraints inℛ\\mathcal\{R\}where they fire, which was the goal set at the end of §[4\.1\.3](https://arxiv.org/html/2605.11117#S4.SS1.SSS3)\.

The procedure that deriveslevel​\(⋅\)\\mathrm\{level\}\(\\cdot\)from𝒯\\mathcal\{T\}andℛ\\mathcal\{R\}is the build\-time pipeline given in Algorithm[2](https://arxiv.org/html/2605.11117#alg2)\(Appendix[A\.1](https://arxiv.org/html/2605.11117#A1.SS1)\): it reduces𝒢A\\mathcal\{G\}\_\{A\}to𝒯\\mathcal\{T\}, identifies the chain set𝒞\\mathcal\{C\}with its structural maps\(ν,ρ\)\(\\nu,\\rho\), expands each rule into the directed edges it induces on𝒞\\mathcal\{C\}, and assigns levels by a monotone fix\-point\. Acyclicity ofHHis checked via Tarjan’s strongly\-connected\-components algorithmTarjan \[[1972](https://arxiv.org/html/2605.11117#bib.bib1)\]: a non\-singleton component is a structural cycle witness and aborts the build\. Each fix\-point pass then extends levels by one edge along the longest path ofHH, so the loop converges inlongest​\-​path​\(H\)\\mathrm\{longest\\text\{\-\}path\}\(H\)passes and the total build cost isO​\(\(\|𝒞\|\+\|E∪Eρ\|\)⋅longest​\-​path​\(H\)\)O\\bigl\(\(\|\\mathcal\{C\}\|\+\|E\\cup E\_\{\\rho\}\|\)\\cdot\\mathrm\{longest\\text\{\-\}path\}\(H\)\\bigr\), dominated by stage 4\.

What the decomposition gives us is more than a rewrite ofπ\\piin fewer parameters\. The cross\-chain couplings collected inℛ\\mathcal\{R\}are exactly the dependencies we cannot afford to lose, and the construction has them all:π~j\\widetilde\{\\pi\}\_\{j\}depends on the lower\-level valuesa<level​\(j\)a\_\{<\\mathrm\{level\}\(j\)\}only through the chains its rule triggers reference, so every coupling has an explicit graphical witness in the dependency structure\. The conditional independencies asserted by the factorisation are read off the absent edges ofHH, which by construction are the chain pairs thatℛ\\mathcal\{R\}does not couple\. WritingpaH​\(j\)=\{i:i→j∈E∪Eρ\}\\mathrm\{pa\}\_\{H\}\(j\)=\\\{\\,i\\,:\\,i\\to j\\in E\\cup E\_\{\\rho\}\\,\\\}for the parents of chainjjinHH, this is the tail\-boundary condition of Verma & Pearl’sII\-map theoremVerma and Pearl \[[1988](https://arxiv.org/html/2605.11117#bib.bib782)\], which yields the following\.

###### Proposition 4\.1\(II\-map of the decision\-level factorisation\)\.

For fixed problemppand fixed repository𝒟\\mathcal\{D\},H=\(𝒞,E∪Eρ\)H=\(\\mathcal\{C\},\\,E\\cup E\_\{\\rho\}\)with parent functionpaH\\mathrm\{pa\}\_\{H\}is anII\-map of the policyπ\(⋅∣p,𝒟\)\\pi\(\\,\\cdot\\,\\mid p,\\mathcal\{D\}\)of Eq\. \([6](https://arxiv.org/html/2605.11117#S4.E6)\), and Eq\. \([6](https://arxiv.org/html/2605.11117#S4.E6)\) is equivalently the canonical Bayesian\-network factorisation onHH\[Pearl,[1988](https://arxiv.org/html/2605.11117#bib.bib781), Ch\. 3 §3\.3\.2\],

π​\(m∣p,𝒟\)=∏j=1kπ¯j​\(aj\|apaH​\(j\),p,𝒟\)\.\\pi\(m\\mid p,\\mathcal\{D\}\)\\;=\\;\\prod\_\{j=1\}^\{k\}\\bar\{\\pi\}\_\{j\}\\bigl\(a\_\{j\}\\bigm\|a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\},\\,p,\\,\\mathcal\{D\}\\bigr\)\.\(7\)

The proof, which verifies the hypotheses of Verma & Pearl’s theorem for the construction of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4), is given in Appendix[A\.1](https://arxiv.org/html/2605.11117#A1.SS1)\. The upshot is that every conditional independence read offHHvia d\-separation holds inπ\(⋅∣p,𝒟\)\\pi\(\\,\\cdot\\,\\mid p,\\mathcal\{D\}\), so no coupling inℛ\\mathcal\{R\}or in the nesting structure of𝒯\\mathcal\{T\}is silently dropped by the per\-chain factorisation\.

Notice that the Bayesian\-network form of Eq\. \([7](https://arxiv.org/html/2605.11117#S4.E7)\) also yields a sharp storage bound\. A dense conditional table for chainjjon its parent context would scale as\|𝒜j\|​∏i∈paH​\(j\)\|𝒜i\|\|\\mathcal\{A\}\_\{j\}\|\\prod\_\{i\\in\\mathrm\{pa\}\_\{H\}\(j\)\}\|\\mathcal\{A\}\_\{i\}\|, and the full joint over the naive method spaceℳ0=𝒜1×⋯×𝒜k\\mathcal\{M\}\_\{0\}=\\mathcal\{A\}\_\{1\}\\times\\cdots\\times\\mathcal\{A\}\_\{k\}as∏j=1k\|𝒜j\|\\prod\_\{j=1\}^\{k\}\|\\mathcal\{A\}\_\{j\}\|, exponential inkk\. GRAFT stores neither: for each chainjjand each internalss\-nodeuu, it keeps a single row over the children ofuu, shared across all parent contexts, and the rule setℛ\\mathcal\{R\}then acts as row masks and overrides whenever triggers fromapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}fire\. Total storage therefore scales as∑j=1k\|𝒜j\|\+\|ℛ\|\\sum\_\{j=1\}^\{k\}\|\\mathcal\{A\}\_\{j\}\|\+\|\\mathcal\{R\}\|rather than∏j=1k\|𝒜j\|\\prod\_\{j=1\}^\{k\}\|\\mathcal\{A\}\_\{j\}\|, reducing the parameter footprint from exponential to linear inkk\(the row\-level construction is taken up in §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)\)\.

#### 4\.1\.5Knowledge graphs and embedding

Notice that the factorisation of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)fixesπ\\pi’s structure but does not yet say how the per\-chain priorsπj\\pi\_\{j\}should be populated for a new problem\. We want the prior to reflect past experience: when a new problempparrives,πj\\pi\_\{j\}should be biased by what worked on problems already solved that resemble it, with closer neighbours weighted more heavily than distant ones\. The categorical paths in𝒯P\\mathcal\{T\}\_\{P\}do not give us a numerical distance directly, so we need to define an embedding of the descriptors of𝒯P\\mathcal\{T\}\_\{P\}to identify which problems are close to each other and to the new problempp\.

Observe that the problem tree𝒯P\\mathcal\{T\}\_\{P\}has nodes that are categorical descriptors \(equation type, dimensionality, boundary conditions, regularity, and so on\), and the action tree𝒯A\\mathcal\{T\}\_\{A\}has nodes that are method decisions \(architecture, optimizer, loss, weighting\)\. Each problempppicks out one root\-to\-leaf pathσP​\(p\)=\(v1,…,vd\)\\sigma\_\{P\}\(p\)=\(v\_\{1\},\\dots,v\_\{d\}\)through𝒯P\\mathcal\{T\}\_\{P\}, and each methodm=\(a1,…,ak\)m=\(a\_\{1\},\\dots,a\_\{k\}\)picks out one root\-to\-leaf path through𝒯A\\mathcal\{T\}\_\{A\}per active chainjj\(inactive chains carryaj=∅ja\_\{j\}=\\varnothing\_\{j\}and contribute none\); we writeσA​\(m\)\\sigma\_\{A\}\(m\)for the union of those chain paths, that isσP\\sigma\_\{P\}records which descriptors a problem instantiates,σA\\sigma\_\{A\}records which decisions a method makes, and together they are the vertex sets that every downstream construction in this section acts on\.

To turn these categorical paths into vectors that can be compared geometrically, each vertexvvof𝒯\\mathcal\{T\}receives a deterministic position in the unit cube via the recursive map

Φ:V→\[0,1\]3,v↦\(x​\(v\),y​\(v\),z​\(v\)\),z​\(v\)=d​\(v\)/D,\\Phi:V\\to\[0,1\]^\{3\},\\qquad v\\mapsto\\bigl\(x\(v\),\\ y\(v\),\\ z\(v\)\\bigr\),\\qquad z\(v\)=d\(v\)/D,\(8\)whereD=maxv∈V⁡d​\(v\)D=\\max\_\{v\\in V\}d\(v\)is the tree depth, soz​\(v\)∈\[0,1\]z\(v\)\\in\[0,1\]and the leaf layer lands onz=1z=1\. The\(x,y\)\(x,y\)coordinates are computed by Algorithm[3](https://arxiv.org/html/2605.11117#alg3)\(Appendix[A\.2](https://arxiv.org/html/2605.11117#A1.SS2)\): the root owns the unit square, every interior vertex sits at the centre of its\(x,y\)\(x,y\)\-rectangle, and its children are slotted into off\-centre sub\-rectangles by splitting alongxxforss\-children \(the mutually\-exclusive picks\) and alongyyforcc\-children \(the joint attributes\)\. For odd\-nngroups the algorithm opensn\+1n\+1slots and drops the middle one, so no child centroid coincides with the parent’s midpoint and the parent’s projection is never shadowed by any descendant\.

We make explicit the one structural property of𝒯\\mathcal\{T\}that the construction relies on: every internal node has children of a single edge type, i\.e\., a node is either subdivided \(ss\-children\) or characterised \(cc\-children\), never both\. The ontology enforces this throughout, so the recursion at everyvvperforms exactly one partition along a single axis, and the children’s sub\-rectangles tilevv’s rectangle without overlap\. Under that assumption,Φ\\Phiis injective:

###### Proposition 4\.2\(Injectivity ofΦ\\Phi\)\.

Let𝒯\\mathcal\{T\}be a finite rooted tree with edge\-type labelsτ\\tausatisfying the uniform\-children structural assumption above\. LetΦ:V​\(𝒯\)→\[0,1\]3\\Phi:V\(\\mathcal\{T\}\)\\to\[0,1\]^\{3\}be the embedding produced by Algorithm[3](https://arxiv.org/html/2605.11117#alg3)\. ThenΦ\\Phiis injective onV​\(𝒯\)V\(\\mathcal\{T\}\), and so is its planar projectionπx​y∘Φ:V​\(𝒯\)→\[0,1\]2\\pi\_\{xy\}\\circ\\Phi:V\(\\mathcal\{T\}\)\\to\[0,1\]^\{2\}\.

The proof is given in Appendix[A\.2](https://arxiv.org/html/2605.11117#A1.SS2)\.

To compare problems through this embedding we discretise the unit cube into aK×K×\(D\+1\)K\\times K\\times\(D\+1\)grid and quantise each vertex into its bin via the boundary\-clamped bin map

βK​\(v\)=\(min⁡\(K−1,⌊K⋅x​\(v\)⌋\),min⁡\(K−1,⌊K⋅y​\(v\)⌋\),d​\(v\)\),\\beta\_\{K\}\(v\)=\\bigl\(\\,\\min\(K\-1,\\ \\lfloor K\\cdot x\(v\)\\rfloor\),\\ \\min\(K\-1,\\ \\lfloor K\\cdot y\(v\)\\rfloor\),\\ d\(v\)\\,\\bigr\),\(9\)so that the planar indices stay in\{0,…,K−1\}\\\{0,\\dots,K\-1\\\}even at the boundaryx​\(v\)=1x\(v\)=1ory​\(v\)=1y\(v\)=1\. The fingerprint of a solved problemPi⊂VP\_\{i\}\\subset Vat resolutionK∈ℕK\\in\\mathbb\{N\}is then

F​\(Pi,K\)=\{βK​\(v\):v∈Pi∩Vkeep\},F\(P\_\{i\},K\)=\\bigl\\\{\\beta\_\{K\}\(v\)\\ :\\ v\\in P\_\{i\}\\cap V\_\{\\text\{keep\}\}\\bigr\\\},\(10\)a finite cell set in the discretised unit cube, equivalently a sparse binary tensor of shapeK×K×\(D\+1\)K\\times K\\times\(D\+1\)\. Pairwise similarity between two fingerprints is the normalised Jaccard

JK​\(Pi,Pj\)=\|F​\(Pi,K\)∩F​\(Pj,K\)\|\|F​\(Pi,K\)∪F​\(Pj,K\)\|∈\[0,1\]\.J\_\{K\}\(P\_\{i\},P\_\{j\}\)=\\frac\{\|F\(P\_\{i\},K\)\\cap F\(P\_\{j\},K\)\|\}\{\|F\(P\_\{i\},K\)\\cup F\(P\_\{j\},K\)\|\}\\in\[0,1\]\.\(11\)The active node setVkeepV\_\{\\text\{keep\}\}controls which edge types contribute: restricting toss\-edges \(the default\) yields a discriminative measure, while including thecc\-backbone instead measures cross\-family resemblance\.

The grid resolutionKKacts as a strictness dial\. LetK⋆K^\{\\star\}denote the smallestKKat whichβK\\beta\_\{K\}is injective on the currentV​\(𝒯\)V\(\\mathcal\{T\}\)\(on the production Numerical problem tree,K⋆=1114K^\{\\star\}=1114\)\. AtK=K⋆K=K^\{\\star\}, distinct nodes never share a cell andJKJ\_\{K\}separates problems down to exact set membership in𝒯P\\mathcal\{T\}\_\{P\}; this is the resolution at which neighbour ranking and policy aggregation are run\. Note that injectivity is asserted atK⋆K^\{\\star\}on the present tree, not for everyK≥K⋆K\\geq K^\{\\star\}: floor binning shifts grid lines asKKvaries, so two nodes separated atK⋆K^\{\\star\}may collide at a coarser or finer resolution\. AtK≪K⋆K\\ll K^\{\\star\}\(in our implementation,K=32K=32\), sibling and nearby\-depth nodes collapse into shared cells, and the fingerprint becomes a fixed\-shape sparse tensor in\{0,1\}K×K×\(D\+1\)\\\{0,1\\\}^\{K\\times K\\times\(D\+1\)\}used for landscape plots and overlay visualisations\. A single scalarKKtherefore interpolates between identity\-preserving retrieval at the operating resolutionK⋆K^\{\\star\}and coarse visualisation at smallKK, with no parallel “strict” and “fuzzy” branches\.

The two vector spaces obtained from𝒯P\\mathcal\{T\}\_\{P\}and𝒯A\\mathcal\{T\}\_\{A\}support four downstream operations used throughout the rest of the paper: a distance between problems \(how similar two PDEs are\), a distance between methods \(how similar two strategies are\), a reward calibration that takes the nearest neighbour’s reward as a baseline, and a landscape visualisation in which problem vectors are projected onto one axis \(via PCA\), method vectors onto another, and points are coloured by reward, error, or time\. Each solved problem becomes a point, and the landscape fills in as ATHENA solves more problems\.

#### 4\.1\.6Policy and prior update

Each chainjjcarries a transition matrixM\(j\)M^\{\(j\)\}, with rowM\(j\)​\(u,⋅\)M^\{\(j\)\}\(u,\\cdot\)the categorical distribution over the children of nodeuuandM\(j\)​\(u,v\)M^\{\(j\)\}\(u,v\)the probability of selecting childvvgiven parentuu\. The chain\-level priorπj∈Δ​\(𝒜j\)\\pi\_\{j\}\\in\\Delta\(\\mathcal\{A\}\_\{j\}\)of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)is the path product of these rows along𝒯A\\mathcal\{T\}\_\{A\}, so populatingπj\\pi\_\{j\}for a new problem reduces to populating the rowsM\(j\)​\(u,⋅\)M^\{\(j\)\}\(u,\\cdot\)\. We writeμ\(j\)\\mu^\{\(j\)\}for the uniform prior on chainjj, withμ\(j\)​\(u,⋅\)\\mu^\{\(j\)\}\(u,\\cdot\)assigning equal probability to every child ofuu\. Rule firings inℛ\\mathcal\{R\}act on these rows via the operatorsℱ,𝒵\\mathcal\{F\},\\mathcal\{Z\}of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4), applied after the data\-driven blend below so that documentation rules cannot be undone by neighbour evidence; the build\-time validator rejects anyzero\_outleaving empty support\.

When a new problempnewp\_\{\\text\{new\}\}arrives, the rows ofM\(j\)M^\{\(j\)\}are re\-estimated from neighbour evidence\. Notice that closeness topnewp\_\{\\text\{new\}\}alone is not enough: a similar problem that was solved badly tells us little about which children ofuuto favour, and a high\-reward problem that is unrelated topnewp\_\{\\text\{new\}\}should not bias the choice either\. We therefore want neighbours that are simultaneously close topnewp\_\{\\text\{new\}\}and were solved well, and the prior should be biased towards their actions in proportion to that joint score\. Concretely, the top\-NnbrN\_\{\\text\{nbr\}\}nearest solved problems are picked byJK⋆J\_\{K^\{\\star\}\}, and each neighbouriicontributes a sigmoid\-gated reward weight

wi=σ​\(sim​\(pnew,pi\)\)⋅rirmax,σ​\(s\)=11\+e−κ​\(s−s0\),\(κ,s0\)=\(7,0\.55\),w\_\{i\}\\;=\\;\\sigma\\bigl\(\\mathrm\{sim\}\(p\_\{\\text\{new\}\},p\_\{i\}\)\\bigr\)\\cdot\\frac\{r\_\{i\}\}\{r\_\{\\max\}\},\\qquad\\sigma\(s\)=\\frac\{1\}\{1\+e^\{\-\\kappa\(s\-s\_\{0\}\)\}\},\\qquad\(\\kappa,\\,s\_\{0\}\)=\(7,\\,0\.55\),\(12\)withsim:=JK⋆\\mathrm\{sim\}:=J\_\{K^\{\\star\}\}onss\-leaves andri/rmax∈\[0,1\]r\_\{i\}/r\_\{\\max\}\\in\[0,1\]\(§[4\.1\.1](https://arxiv.org/html/2605.11117#S4.SS1.SSS1)\)\. Each neighbouriithen contributes a per\-node distributionSiS\_\{i\}over the children of every internalu∈𝒯Au\\in\\mathcal\{T\}\_\{A\}, set to a one\-hot vote on rows alongσA​\(mi\)\\sigma\_\{A\}\(m\_\{i\}\)and to the prior on rows it did not visit,

Si​\(u,⋅\)=\{δmi​\(u\),u∈σA​\(mi\),μ​\(u,⋅\),u∉σA​\(mi\),S\_\{i\}\(u,\\cdot\)=\\begin\{cases\}\\delta\_\{m\_\{i\}\(u\)\},&u\\in\\sigma\_\{A\}\(m\_\{i\}\),\\\\\[2\.0pt\] \\mu\(u,\\cdot\),&u\\notin\\sigma\_\{A\}\(m\_\{i\}\),\\end\{cases\}\(13\)whereδv\\delta\_\{v\}is the one\-hot row placing all mass on childvvandmi​\(u\)m\_\{i\}\(u\)is the child ofuuchosen by neighbourii’s method along its root\-to\-leaf path\. The data\-driven estimate is thewiw\_\{i\}\-weighted average of these contributions across the full top\-NnbrN\_\{\\text\{nbr\}\}set, written here with the path\-on / path\-off split made explicit,

Mdata​\(u,⋅\)=1Wtot​\[∑i:u∈σA​\(mi\)wi​δmi​\(u\)\+∑i:u∉σA​\(mi\)wi​μ​\(u,⋅\)\],Wtot=∑i=1Nnbrwi,M\_\{\\text\{data\}\}\(u,\\cdot\)=\\frac\{1\}\{W\_\{\\text\{tot\}\}\}\\Biggl\[\\sum\_\{i\\,:\\,u\\in\\sigma\_\{A\}\(m\_\{i\}\)\}w\_\{i\}\\,\\delta\_\{m\_\{i\}\(u\)\}\\;\+\\;\\sum\_\{i\\,:\\,u\\notin\\sigma\_\{A\}\(m\_\{i\}\)\}w\_\{i\}\\,\\mu\(u,\\cdot\)\\Biggr\],\\qquad W\_\{\\text\{tot\}\}=\\sum\_\{i=1\}^\{N\_\{\\text\{nbr\}\}\}w\_\{i\},\(14\)with the trivial fallbackMdata=μM\_\{\\text\{data\}\}=\\muwhenWtot=0W\_\{\\text\{tot\}\}=0\. On rows every neighbour visits, the second sum is empty and the average reduces to awiw\_\{i\}\-weighted vote among their picks; on rows no neighbour visits, the first sum is empty and the average collapses toμ\\mu; intermediate rows interpolate, with shrinkage towardμ\\mugrowing as the visiting weight share∑i:u∈σA​\(mi\)wi/Wtot\\sum\_\{i:\\,u\\in\\sigma\_\{A\}\(m\_\{i\}\)\}w\_\{i\}\\,/\\,W\_\{\\text\{tot\}\}decreases\. The final policy is a row\-wise convex combination ofMdataM\_\{\\text\{data\}\}withμ\\mu,

M=W¯⋅Mdata\+\(1−W¯\)⋅μ,W¯=clip​\(WtotNeff,0,1\),Neff=\|\{i:wi\>0\}\|,M=\\bar\{W\}\\cdot M\_\{\\text\{data\}\}\+\(1\-\\bar\{W\}\)\\cdot\\mu,\\qquad\\bar\{W\}=\\mathrm\{clip\}\\\!\\left\(\\frac\{W\_\{\\text\{tot\}\}\}\{N\_\{\\text\{eff\}\}\},\\,0,\\,1\\right\),\\qquad N\_\{\\text\{eff\}\}=\\bigl\|\\\{i:w\_\{i\}\>0\\\}\\bigr\|,\(15\)where the mixing weightW¯\\bar\{W\}is the average confidence among neighbours that contribute non\-trivial evidence \(with the conventionW¯=0\\bar\{W\}=0whenNeff=0N\_\{\\text\{eff\}\}=0\)Hoetinget al\.\[[1999](https://arxiv.org/html/2605.11117#bib.bib783)\]\. HighW¯\\bar\{W\}trusts the data\-driven estimate, lowW¯\\bar\{W\}falls back to the prior, andW¯=0\\bar\{W\}=0degeneratesMMcleanly toμ\\mu\. Rule operators are then applied on top ofMM, so any forced or forbidden support encoded inℛ\\mathcal\{R\}overrides the blended estimate\. Algorithm[5](https://arxiv.org/html/2605.11117#alg5)\(Appendix[A\.4](https://arxiv.org/html/2605.11117#A1.SS4)\) bundles the full procedure\.

After solvingpnewp\_\{\\text\{new\}\}and observing\(m,r,O\)\(m,r,O\), the tuple\(pnew,m,r,O\)\(p\_\{\\text\{new\}\},m,r,O\)is stored in𝒟\\mathcal\{D\}, so future queries nearpnewp\_\{\\text\{new\}\}may include it in their top\-NnbrN\_\{\\text\{nbr\}\}aggregation and the policy is updated without explicit retraining\. When the action graph itself is expanded, new nodes inherit priors from their siblings and removed nodes redistribute mass back to siblings; the repository𝒟\\mathcal\{D\}therefore grows monotonically in stored observations\. The local prior used for any given query may still shift if the top\-NnbrN\_\{\\text\{nbr\}\}neighbour set changes, so monotone growth of𝒟\\mathcal\{D\}does not imply monotone improvement of the policy at every individual problem\.

#### 4\.1\.7From prior to agent

Notice thatπ​\(m∣p\)\\pi\(m\\mid p\)in Eq\. \([7](https://arxiv.org/html/2605.11117#S4.E7)\), with rowsM\(j\)​\(u,⋅\)M^\{\(j\)\}\(u,\\cdot\)populated by §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6), is built from𝒢A\\mathcal\{G\}\_\{A\}alone: those rows inherit problem context only through the policy update of §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6), which leans on neighbour problems already solved\. Two gaps remain at sampling time\. First, the present problempphas its own descriptors in𝒯P\\mathcal\{T\}\_\{P\}\(dimensionality, boundary type, regularity, and so on\) that the prior cannot read directly; a single draw fromπ\(⋅∣p\)\\pi\(\\cdot\\mid p\)ignores any structure ofppthat the neighbour aggregation has not already absorbed\. Second, the leaves of𝒯A\\mathcal\{T\}\_\{A\}are categorical labels, but the actual specification of an action is not\. A choice such as “MLP” carries continuous knobs \(learning rate, width, depth\) and structural sub\-decisions \(variable ordering for KAN\-style models, layer\-wise loss weighting, implementation prescriptions\) that are virtually non\-finite and that no categorical sampler over𝒯A\\mathcal\{T\}\_\{A\}can produce\.

Observe that closing both gaps withπ\\pialone is not realistic\. Enumerating every continuous knob inside the tree is impossible, and waiting for the row prior to sharpen enough that any sample from it is automatically calibrated to the present problem is expensive in practice: a flat prior at small\|𝒟\|\|\\mathcal\{D\}\|has to rediscover by trial what the literature already records\. ATHENA’s inviscid\-Burgers runs illustrate this cost: several attempts were needed before the row mass concentrated on the conservative formulation that is, in retrospect, a textbook choice for that family of equations\. The same cost would recur on every new problem class until the corpus is large enough to cover it\.

We close both gaps with two agents working in tandem on the priorπ\(⋅∣p\)\\pi\(\\cdot\\mid p\): a proposer that samples fromπ\\piat each level and a critic that audits the sample\. The pair traversesHHlevel by level\. At each level, the proposer is shown the active chains’ children together with their current row probabilitiesM\(j\)​\(u,⋅\)M^\{\(j\)\}\(u,\\cdot\), the hints attached to those children, and the descriptors ofppread from𝒯P\\mathcal\{T\}\_\{P\}, and it emits one actionaja\_\{j\}per active chain, realising any sub\-leaf knobs \(learning rate, width, depth, variable orderings, loss weightings\) inline\. The proposer’s default is to follow the probabilities; it deviates only when a hint is more informative than the prior, for instance steering “problem is periodic” toward a periodic embedding even if row mass favours a generic choice\. Once a level’s picks are fixed, the rule operators of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)bite deterministically on the next level’s rows —ℱ\\mathcal\{F\}forcing support where a triggered rule says “must”,𝒵\\mathcal\{Z\}zeroing it where the rule says “forbidden” — soapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}is fully resolved before chainjjis sampled, matching the conditioning structure of Eq\.[7](https://arxiv.org/html/2605.11117#S4.E7)\. The critic then verifies that the proposer’s emission respected both the probabilities and the hints; rejections trigger a re\-proposal at the same level, and the next level is taken up only after the current level passesJhaet al\.\[[2023](https://arxiv.org/html/2605.11117#bib.bib790)\]\.

With the agent in the loop, the realised method distribution is notπ\\pibut an agent\-induced distributionPagentP\_\{\\text\{agent\}\}overℳ\\mathcal\{M\}, sampled chain by chain in the topological order ofHH\. To certify thatHHremains anII\-map ofPagentP\_\{\\text\{agent\}\}, we make precise the role of the hints\.

###### Assumption 4\.3\(Locality of the proposer–critic pair\)\.

For every chainjj, the hints presented to the proposer are a deterministic function

hintsj=hj​\(j,apaH​\(j\)\)\\mathrm\{hints\}\_\{j\}\\;=\\;h\_\{j\}\\\!\\bigl\(j,\\,a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}\\bigr\)of the chain identity and the values selected on its parents inHH, and the proposer’s policy at chainjjdepends on the predecessor history only throughapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}andhintsj\\mathrm\{hints\}\_\{j\}\. The critic’s acceptance event at chainjjis likewise a function only of the candidateaja\_\{j\}, the parent picksapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}, the local rowM\(j\)​\(u,⋅\)M^\{\(j\)\}\(u,\\cdot\), andhintsj\\mathrm\{hints\}\_\{j\}, and does not depend on non\-parent predecessor history\. Finally, when the strategist consults the short\-term history𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}to avoid revisiting failed configurations, the filter applied at chainjjdepends only on the per\-chain projection𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\(j\)=\{\(ai,j,ai,paH​\(j\),Ri\):i<n\}\\mathsf\{History\}\_\{n\}^\{\(j\)\}=\\\{\(a\_\{i,j\},a\_\{i,\\mathrm\{pa\}\_\{H\}\(j\)\},R\_\{i\}\):i<n\\\}, that is, on prior values at chainjjand itsHH\-parents together with their realised rewards\.

###### Proposition 4\.4\(Per\-stepII\-map under the agent\)\.

For fixed problempp, fixed repository𝒟\\mathcal\{D\}, and fixed within\-trial history𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}at iterationnn, under Assumption[4\.3](https://arxiv.org/html/2605.11117#S4.Thmtheorem3),HHwith parent functionpaH\\mathrm\{pa\}\_\{H\}is anII\-map of the per\-step distributionPagent\(⋅∣p,𝒟,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\)P\_\{\\text\{agent\}\}\(\\,\\cdot\\,\\mid p,\\mathcal\{D\},\\mathsf\{History\}\_\{n\}\), and

Pagent​\(m∣p,𝒟,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\)=∏j=1kPagent​\(aj\|apaH​\(j\),p,𝒟,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\(j\)\)\.P\_\{\\text\{agent\}\}\(m\\mid p,\\mathcal\{D\},\\mathsf\{History\}\_\{n\}\)\\;=\\;\\prod\_\{j=1\}^\{k\}P\_\{\\text\{agent\}\}\\\!\\bigl\(a\_\{j\}\\bigm\|a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\},\\,p,\\,\\mathcal\{D\},\\,\\mathsf\{History\}\_\{n\}^\{\(j\)\}\\bigr\)\.\(16\)

The proof, which checks the tail\-boundary condition of Theorem[A\.2](https://arxiv.org/html/2605.11117#A1.Thmtheorem2)forPagentP\_\{\\text\{agent\}\}, is given in Appendix[A\.1](https://arxiv.org/html/2605.11117#A1.SS1)\.

Beyond sampling,𝒯A\\mathcal\{T\}\_\{A\}itself is flexible: once a problem is solved, the user or an agent may grow𝒢A\\mathcal\{G\}\_\{A\}\(and thereby𝒯A\\mathcal\{T\}\_\{A\}\) by attaching new actions or by adding rules toℛ\\mathcal\{R\}\. New nodes inherit priors from siblings \(§[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)\) and𝒟\\mathcal\{D\}persists across the expansion\. The extension re\-enters Algorithm[2](https://arxiv.org/html/2605.11117#alg2)with the new pair\(𝒯′,ℛ′\)\(\\mathcal\{T\}^\{\\prime\},\\mathcal\{R\}^\{\\prime\}\), and the recompiledH′H^\{\\prime\}inherits theII\-map certification above as long asH′H^\{\\prime\}is acyclic, which the build\-time check enforces\.

This places the architecture in the lineage of expert systems and earlier symbolic\-AI searchQuillan \[[1966](https://arxiv.org/html/2605.11117#bib.bib810)\], Newell and Simon \[[1956](https://arxiv.org/html/2605.11117#bib.bib811)\], Newell \[[1963](https://arxiv.org/html/2605.11117#bib.bib812)\], Hartet al\.\[[1968](https://arxiv.org/html/2605.11117#bib.bib813)\], Wooldridge \[[2002](https://arxiv.org/html/2605.11117#bib.bib814)\]but defends against their classical limitations on four counts\. The rule setℛ\\mathcal\{R\}comes from documentation prose rather than handcrafted code, so authoring scales with the literature and updates with it\. The priorπ\\piis data\-driven and sharpens with\|𝒟\|\|\\mathcal\{D\}\|, so the system learns rather than freezing at its initial encoding\. Hints are enrichments rather than load\-bearing logic: when a hint is missing or wrong, the system still samples throughπ\\piandℛ\\mathcal\{R\}, paying only an exploration cost\. And sub\-leaf realisation, which classical expert systems could not represent at all, lives in the agent rather than in the symbolic layer\. As\|𝒟\|\|\\mathcal\{D\}\|grows, the marginal contribution of the hints decreases and the prior dominates; at small\|𝒟\|\|\\mathcal\{D\}\|, the hints carry the load\. The handoff is graceful, not switched\.

#### 4\.1\.8The closed loop: GRAFT\-ATHENA cycle

The system carries two memories\. The repository𝒟\\mathcal\{D\}is the long\-term store, persistent across problems and the only object the prior is compiled from\. The trial history𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}is the short\-term store, scoped to the current problempnewp\_\{\\text\{new\}\}and used by the strategist to avoid re\-sampling configurations that have already failed within the trial; at trial end,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}is folded into𝒟\\mathcal\{D\}entry\-by\-entry and reset for the next problem\. In Pearl’s vocabulary,\(𝒟,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\)\(\\mathcal\{D\},\\mathsf\{History\}\_\{n\}\)is the beginning of an abduction substrate for retrospective queries: each recorded tuple\(pi,mi,Oi,ri\)\(p\_\{i\},m\_\{i\},O\_\{i\},r\_\{i\}\)provides the evidence such a query would condition on, but full abduction would require richer trace logging of implementation edits, solver seeds, proposal state, and other exogenous variables\.

A trial proceeds as a sequence of stepsn=0,1,…n=0,1,\\dots\. At stepnn, the system maintains a context𝖢𝗈𝗇𝗍𝖾𝗑𝗍n\\mathsf\{Context\}\_\{n\}\(the problem statementpnewp\_\{\\text\{new\}\}, the current code stateSnS\_\{n\}, and the short\-term history𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}\); the strategist agent, guided by the closed\-loop policyΠ​\(An∣𝖢𝗈𝗇𝗍𝖾𝗑𝗍n\)\\Pi\(A\_\{n\}\\mid\\mathsf\{Context\}\_\{n\}\), selects an actionAn=mn=\(an,1,…,an,k\)∈ℳA\_\{n\}=m\_\{n\}=\(a\_\{n,1\},\\dots,a\_\{n,k\}\)\\in\\mathcal\{M\}, and the system steps according to

Sn\+1=ℐ​\(An,Sn\),On=𝖤𝗑𝖾𝖼​\(Sn\+1\),Rn=R​\(On\),S\_\{n\+1\}=\\mathcal\{I\}\(A\_\{n\},S\_\{n\}\),\\qquad O\_\{n\}=\\mathsf\{Exec\}\(S\_\{n\+1\}\),\\qquad R\_\{n\}=R\(O\_\{n\}\),\(17\)whereℐ\\mathcal\{I\}is the implementation operator \(realisesAnA\_\{n\}as code edits applied toSnS\_\{n\}\),𝖤𝗑𝖾𝖼\\mathsf\{Exec\}is the execution operator \(runs the code, returns observations\), andRRscores observations into a scalar reward\. In Pearl’s structural\-causal vocabulary, conditional on the problem instancepnewp\_\{\\text\{new\}\}, the code state, solver settings, seeds, and runtime environment,𝖤𝗑𝖾𝖼\\mathsf\{Exec\}plays the role of the structural map carrying configured code states to their observable consequences\. The agent’s pickAnA\_\{n\}is therefore an intervention on the executable solver state, not a passive observation\. The chain dependency graphHHshould be read more narrowly: it is a policy\-dependencyII\-map over action choices \(Proposition[4\.4](https://arxiv.org/html/2605.11117#S4.Thmtheorem4)\), while the causal intervention occurs when the selected method is realised and executed through𝖤𝗑𝖾𝖼\\mathsf\{Exec\}\. The short\-term history grows as𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\+1=𝖧𝗂𝗌𝗍𝗈𝗋𝗒n∪\{\(An,On,Rn\)\}\\mathsf\{History\}\_\{n\+1\}=\\mathsf\{History\}\_\{n\}\\cup\\\{\(A\_\{n\},O\_\{n\},R\_\{n\}\)\\\}\. The closed\-loop policyΠ\\Pidraws through the rows of the compiled priorπ\(⋅∣pnew,𝒟\)\\pi\(\\cdot\\mid p\_\{\\text\{new\}\},\\mathcal\{D\}\), which is held fixed for the duration of the trial; the proposer’s and critic’s selection within each row is filtered by𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}so that configurations already tried onpnewp\_\{\\text\{new\}\}are avoided\. Sampling itself proceeds via the explicit factored distribution of Eq\.[16](https://arxiv.org/html/2605.11117#S4.E16): the agent traverses the chain DAG in topological order and, at each chainjj, is guided by the rowM\(j\)​\(u,⋅\)M^\{\(j\)\}\(u,\\cdot\)together with the within\-trial filter from𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}\. Graph\-encoded hints baked into𝒯A\\mathcal\{T\}\_\{A\}at construction \(the dependency rulesRRof §[4\.1\.3](https://arxiv.org/html/2605.11117#S4.SS1.SSS3)and the grading prose attached to each node\) enter𝖢𝗈𝗇𝗍𝖾𝗑𝗍n\\mathsf\{Context\}\_\{n\}alongside the runtime state, giving the agent direct read access to the structural constraints at every step rather than re\-deriving them from the policy distribution\.

Given a new problempnewp\_\{\\text\{new\}\}, the cycle proceeds as in Algorithm[1](https://arxiv.org/html/2605.11117#alg1): GRAFT compiles the prior, ATHENA runs the inner trial loop on the sampled method, and the resulting trace appends to𝒟\\mathcal\{D\}at every iteration so the repository grows monotonically\. Becausewi=σ​\(sim\)⋅ri/rmaxw\_\{i\}=\\sigma\(\\mathrm\{sim\}\)\\cdot r\_\{i\}/r\_\{\\max\}\(Eq\.[12](https://arxiv.org/html/2605.11117#S4.E12)\) is non\-negative throughout, low\-reward attempts receive low reinforcement rather than explicit negative evidence, and successful strategies dominate the weighted aggregation while failed ones merely enlarge the support over which it operates\.

Input:new problem

pnewp\_\{\\text\{new\}\}; repository

𝒟\\mathcal\{D\}; rule set

ℛ\\mathcal\{R\}; budget

TT\.

Output:converged method

m⋆m^\{\\star\}; updated repository

𝒟\\mathcal\{D\}\.

1exbuild

π\(⋅∣pnew,𝒟\)\\pi\(\\cdot\\mid p\_\{\\text\{new\}\},\\mathcal\{D\}\)via Algorithm[5](https://arxiv.org/html/2605.11117#alg5)\(rows

M\(j\)M^\{\(j\)\}, then rule operators\)

n←0n\\leftarrow 0; initialise

S0S\_\{0\},

𝖧𝗂𝗌𝗍𝗈𝗋𝗒0\\mathsf\{History\}\_\{0\}
repeat

sample method

mn=\(an,1,…,an,k\)m\_\{n\}=\(a\_\{n,1\},\\dots,a\_\{n,k\}\)from the closed\-loop policy

Π\(⋅∣pnew,𝒟,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n,Sn\)\\Pi\(\\,\\cdot\\,\\mid p\_\{\\text\{new\}\},\\mathcal\{D\},\\mathsf\{History\}\_\{n\},S\_\{n\}\), drawing through the rows of

π\(⋅∣pnew,𝒟\)\\pi\(\\cdot\\mid p\_\{\\text\{new\}\},\\mathcal\{D\}\)in topological order on

HHwith the proposer/critic of §[4\.1\.7](https://arxiv.org/html/2605.11117#S4.SS1.SSS7)

An←mnA\_\{n\}\\leftarrow m\_\{n\}
Sn\+1←ℐ​\(An,Sn\)S\_\{n\+1\}\\leftarrow\\mathcal\{I\}\(A\_\{n\},S\_\{n\}\);

On←𝖤𝗑𝖾𝖼​\(Sn\+1\)O\_\{n\}\\leftarrow\\mathsf\{Exec\}\(S\_\{n\+1\}\);

Rn←R​\(On\)R\_\{n\}\\leftarrow R\(O\_\{n\}\)
//Eq\.[17](https://arxiv.org/html/2605.11117#S4.E17)

𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\+1←𝖧𝗂𝗌𝗍𝗈𝗋𝗒n∪\{\(An,On,Rn\)\}\\mathsf\{History\}\_\{n\+1\}\\leftarrow\\mathsf\{History\}\_\{n\}\\cup\\\{\(A\_\{n\},O\_\{n\},R\_\{n\}\)\\\}
𝒟←𝒟∪\{\(pnew,mn,On,Rn\)\}\\mathcal\{D\}\\leftarrow\\mathcal\{D\}\\cup\\\{\(p\_\{\\text\{new\}\},m\_\{n\},O\_\{n\},R\_\{n\}\)\\\}
advisor proposes node\-level edits to

mnm\_\{n\}from

\(On,Rn\)\(O\_\{n\},R\_\{n\}\)
n←n\+1n\\leftarrow n\+1
until*convergenceorn=Tn=T*

return

m⋆←arg⁡maxt≤n⁡Rtm^\{\\star\}\\leftarrow\\arg\\max\_\{t\\leq n\}R\_\{t\}, updated

𝒟\\mathcal\{D\}

Algorithm 1GRAFT\-ATHENA closed loop on a new problempnewp\_\{\\text\{new\}\}\.The policy evolves across three timescales\. Within a problem, the compiled priorπ\(⋅∣pnew,𝒟\)\\pi\(\\cdot\\mid p\_\{\\text\{new\}\},\\mathcal\{D\}\)is held fixed, and adaptation occurs through the short\-term history filter, advisor corrections, and code\-state updates: observed low\-reward configurations enter𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}and are avoided on later samples from the same trial, while the advisor proposes node\-level edits tomn−1m\_\{n\-1\}on𝒯A\\mathcal\{T\}\_\{A\}from\(On−1,Rn−1\)\(O\_\{n\-1\},R\_\{n\-1\}\)at iterationsn≥1n\\geq 1\. Across problems, every iteration, successful or failed, persists into𝒟\\mathcal\{D\}, so the next problem’s prior is built from a repository containing weakly more observations than before through the sigmoid\-gated, reward\-weighted blend of Eq\.[15](https://arxiv.org/html/2605.11117#S4.E15)\. At the schema\-growth timescale, when an agent proposes a method outside𝒯A\\mathcal\{T\}\_\{A\}’s current support,𝒢A\\mathcal\{G\}\_\{A\}is grown, the spanning\-tree projection regrows𝒯A\\mathcal\{T\}\_\{A\}, andπ\\pi’s domain literally expands; new nodes inherit priors from siblings, and𝖧𝗂𝗌𝗍𝗈𝗋𝗒\\mathsf\{History\}and𝒟\\mathcal\{D\}persist across the expansion\.

The three Pearl ingredients are therefore scaffolded, though not yet fully exercised\. The chain dependency graphHHon𝒢A\\mathcal\{G\}\_\{A\}is a policy\-dependencyII\-map of the agent\-induced action distribution \(Proposition[4\.4](https://arxiv.org/html/2605.11117#S4.Thmtheorem4)\), the operator𝖤𝗑𝖾𝖼\\mathsf\{Exec\}is the structural map carrying configured code states to observations, and\(𝒟,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\)\(\\mathcal\{D\},\\mathsf\{History\}\_\{n\}\)together with the per\-trial trace\(An,On,Rn\)\(A\_\{n\},O\_\{n\},R\_\{n\}\)provides the evidence on which future retrospective queries would condition\. A query of the form “had the agent descended the alternative branch at decision nodev∈𝒯Av\\in\\mathcal\{T\}\_\{A\}, what would the outcome have been?” is therefore formulable in principle on this scaffold via the standard three\-step recipe \(abduction, action, prediction\)Pearl \[[2018](https://arxiv.org/html/2605.11117#bib.bib17),[2019](https://arxiv.org/html/2605.11117#bib.bib9)\]: it would require abducting the relevant exogenous state from a sufficiently rich recorded trace, intervening on𝒯A\\mathcal\{T\}\_\{A\}by forcing the alternative branch atvv, and predicting by re\-evaluating𝖤𝗑𝖾𝖼\\mathsf\{Exec\}under that abducted state\. Notice that the Pearl machinery rests on𝒢A\\mathcal\{G\}\_\{A\}alone: the problem graph𝒢P\\mathcal\{G\}\_\{P\}enters the construction as a structured descriptor space whose fingerprintFF\(§[4\.1\.5](https://arxiv.org/html/2605.11117#S4.SS1.SSS5)\) identifies the conditioning eventpnewp\_\{\\text\{new\}\}, but its taxonomic edges encode vocabulary relationships rather than mechanistic ones, and the system never intervenes on a problem descriptor\. In this work we exercise the scaffold at Levels 1 and 2, namely associational priors \(§[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)\) and the interventional closed loop above; full counterfactual identification would require richer trace logging of proposal state, solver seeds, implementation edits, and other exogenous variables, and is left to follow\-up\.

### 4\.2GRAFT\-ATHENA teams

#### 4\.2\.1Formalization Team

GRAFT\-ATHENA’s performance depends heavily on the quality of the problem statement, so problems that are defined properly are strongly preferredRoggeveenet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib793)\]\. To address this gap, the first team of the pipeline is the Formalization team \(Fig\.[1](https://arxiv.org/html/2605.11117#S1.F1)C\), which interacts with the user and is composed of four agents acting in sequence\. The first is a formalization agent that extracts the information needed to specify the problem \(e\.g\., the specific PDE, the boundary and initial conditions, whether the problem is inverse, whether data are available and whether they carry noise, and the acceptable assumptions\), and defines it in a form suitable for the downstream spine\.

Based on this information, a subsequent analysis takes place\. A second agent reads the formalized user request and looks for two kinds of features, namely exact solutions and simplifications, and proposes several candidate alternatives in each category\. A third agent then ranks these proposals against a set of constraints, namely dimension reduction, nonlinear\-term reduction, regularity constraints, the cost of the boundary conditions, the implementation cost, and composability, while interacting with the user to surface a modified user request whenever a change is needed\. The two agents iterate as a proposer\-critic pair until they converge on a winning candidate\. Notice that not every problem can or should be simplified, so the ranker may also conclude that none of the alternatives is suitable, in which case the original statement is forwarded unchanged\. Once a winner has been selected, the ranker rederives it step by step and engages a fourth agent that audits each step; the two go back and forth in a second proposer\-critic loop until the derivation is clean and consistent with the original statement\.

After the user request has been formalized and, where applicable, simplified and cleanly rederived, the same fourth agent performs a well\-posedness analysis on the resulting problem\. If existence, uniqueness, or stability cannot be established, the agent interacts with the user to gather the missing information and proposes additional constraints that render the problem conditionally well\-posed before it enters the encode\-select\-solve spine\.

### 4\.3PIML action\-tree construction

Unlike the Trixi\.jl and Nektar\+\+ branches of𝒯A\\mathcal\{T\}\_\{A\}, which are constructed by ingesting solver documentation, the PIML branch was seeded manually from the PINN/PIML and numerical\-analysis literature\. This subsection records that provenance: it specifies which action families were admitted, which dependencies were encoded as cross\-rules, and which literature\-backed hints were attached to each node before GRAFT projected the seed graph into𝒯A\\mathcal\{T\}\_\{A\}\. The goal is not a catalogue of promptable tricks, but an action vocabulary in which representation choices, residual objectives, optimizers, PDE reformulations, and numerical discretisations enter as inspectable leaves and rules, so that the proposer\-critic loop can sample, audit, and revise them rather than emit them as free text\.

##### Representation scaffold\.

The representation branch follows the standard approximation\-theoretic view that a neural solver chooses a function class in which the PDE solution is representedCybenko \[[1989](https://arxiv.org/html/2605.11117#bib.bib246)\], Horniket al\.\[[1989](https://arxiv.org/html/2605.11117#bib.bib247)\], Raissiet al\.\[[2019](https://arxiv.org/html/2605.11117#bib.bib65)\], Karniadakiset al\.\[[2021](https://arxiv.org/html/2605.11117#bib.bib64)\]\. This scaffold seeds leaves for MLP backbones, input embeddings, Fourier features, and hard output ansatzes that absorb boundary or initial conditions structurally, with the spectral\-bias view of Fourier feature networks supplying the diagnostic that flags multi\-scale targets where plain MLPs underfit high\-frequency contentWanget al\.\[[2020](https://arxiv.org/html/2605.11117#bib.bib180),[2021b](https://arxiv.org/html/2605.11117#bib.bib138)\]\. Additional approximation families enter as optional backbone leaves rather than as load\-bearing assumptions, covering Kolmogorov\-Arnold networks and their smooth\-superposition refinementKolmogorov \[[1957](https://arxiv.org/html/2605.11117#bib.bib719)\], Liuet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib199)\], Songet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib505)\], the KKAN variant designed for PIML representationToscanoet al\.\[[2025d](https://arxiv.org/html/2605.11117#bib.bib49)\], expressivity and spectral\-bias bounds that delimit when KAN bases recover or improve on MLP behaviourWanget al\.\[[2025b](https://arxiv.org/html/2605.11117#bib.bib499)\], and initialization schemes that stabilise their trainingRigaset al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib707)\]\. Compatibility hints link periodicity, smoothness, dimensionality, and prescribed boundary data to the representations most likely to be admissible for them, with the FAIR cross\-architecture comparison and a recent KAN review acting as the umbrella references against which these leaves are admittedShuklaet al\.\[[2024](https://arxiv.org/html/2605.11117#bib.bib182)\], Faroughiet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib708)\]\.

##### Residual\-objective scaffold\.

The loss branch treats a PIML objective as residual minimisation under a chosen weighting and sampling measure on the domain\. Uniform residual mean\-squared error, self\-adaptive weighting, NTK and gradient\-balancing diagnostics, residual\-based attention \(RBA, in which high\-residual evaluations receive larger weights or sampling probability\), and vector RBA \(vRBA, its multi\-index extension for residuals indexed by field component, equation, mode, or time slab\) live as alternatives within a single action family rather than as independent tricksWanget al\.\[[2022](https://arxiv.org/html/2605.11117#bib.bib472),[2021a](https://arxiv.org/html/2605.11117#bib.bib130)\], McClenny and Braga\-Neto \[[2023](https://arxiv.org/html/2605.11117#bib.bib203)\], Wuet al\.\[[2023b](https://arxiv.org/html/2605.11117#bib.bib475)\], Anagnostopouloset al\.\[[2024](https://arxiv.org/html/2605.11117#bib.bib195),[2025](https://arxiv.org/html/2605.11117#bib.bib93)\]\. Noise\-aware likelihoods enter as a parallel leaf in the same family, with a negative\-log\-likelihood objective that learns per\-point noise variance jointly with the field and downweights observations whose residual is consistent with their noise levelToscanoet al\.\[[2024](https://arxiv.org/html/2605.11117#bib.bib194)\]\. Across these alternatives, the variational residual\-adaptivity framework supplies the organising principle: changing the sampling, weighting, or likelihood rule changes the effective norm being optimised, so each leaf carries compatibility hints that match it to stiff, localised, noisy, or multi\-component residualsToscanoet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib23)\]\.

##### Optimizer scaffold\.

The optimizer branch records the update rule, curvature model, line\-search or step\-size policy, and training schedule as separate actions on𝒯A\\mathcal\{T\}\_\{A\}\. First\-order methods such as Adam, quasi\-Newton methods such as L\-BFGS, and SSBroyden\-style curvature\-aware updates therefore occupy comparable positions in the tree rather than competing inside a single decisionKingma and Ba \[[2014](https://arxiv.org/html/2605.11117#bib.bib484)\], Liu and Nocedal \[[1989](https://arxiv.org/html/2605.11117#bib.bib200)\], Nocedal and Wright \[[1999](https://arxiv.org/html/2605.11117#bib.bib201)\], Urbánet al\.\[[2024](https://arxiv.org/html/2605.11117#bib.bib496)\]\. Hints connect stiff or ill\-conditioned PINN regimes to second\-order or line\-search choices, and tie schedule edits and warm\-restart strategies to the curvature regime in which they tend to be productiveKiyaniet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib38)\], Jniniet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib11)\]\.

##### PDE\-reformulation scaffold\.

The PDE branch exposes transformations of the mathematical problem itself: exact or approximate reductions, hard boundary or initial\-condition ansatzes, conservation\-law reparametrisations such as stream\-function or vector\-potential forms, continuation schedules in physical or numerical parameters, and regularisers introduced to close identifiability gapsRaissiet al\.\[[2019](https://arxiv.org/html/2605.11117#bib.bib65)\], Karniadakiset al\.\[[2021](https://arxiv.org/html/2605.11117#bib.bib64)\], Evans \[[2022](https://arxiv.org/html/2605.11117#bib.bib31)\]\. The inverse\-problem lineage enters this branch as a distinct family of moves, since hidden fields are encoded directly as network outputs and inferred through the residual itselfRaissiet al\.\[[2020](https://arxiv.org/html/2605.11117#bib.bib164)\], with downstream extensions covering in\-vivo brain flow with simultaneous pressure and permeability inferenceToscanoet al\.\[[2025b](https://arxiv.org/html/2605.11117#bib.bib89)\], Vaeziet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib92)\], vorticity reformulations that eliminate the pressure unknown and use theoretical profiles as sequential\-training scaffoldsToscanoet al\.\[[2025c](https://arxiv.org/html/2605.11117#bib.bib41)\], multi\-balance reactor inference under coupled flow, material, and energy residualsWuet al\.\[[2025](https://arxiv.org/html/2605.11117#bib.bib90)\], and operator\-learning counterparts that reuse the same hidden\-field inference under a data\-efficient surrogateWanget al\.\[[2026b](https://arxiv.org/html/2605.11117#bib.bib91)\]\. This scaffold is what allows the Formalization team \(§[4\.2\.1](https://arxiv.org/html/2605.11117#S4.SS2.SSS1)\) to treat well\-posedness, constraint enforcement, and PDE simplification as selectable, auditable actions before implementation rather than as silent prompt\-level reformulations\. The perivascular inverse problem \(§[2\.7](https://arxiv.org/html/2605.11117#S2.SS7)\) exercises this branch when mass conservation is encoded structurally through a vector\-potential formulation and an unobservable slab mode is regularised; the spectral Burgers case \(§[2\.8](https://arxiv.org/html/2605.11117#S2.SS8)\) exercises it when the Fourier\-Galerkin reduction, hard initial\-condition ansatz, parity fold, and dealiased nonlinear projection are promoted into action\-tree leaves rather than left implicit in codeToscanoet al\.\[[2026](https://arxiv.org/html/2605.11117#bib.bib23)\]\.

##### Numerical\-method scaffold\.

The numerical branch organises solver choices by PDE character and expected solution regularity\. Smooth periodic problems activate spectral and Fourier\-Galerkin hintsShenet al\.\[[2011](https://arxiv.org/html/2605.11117#bib.bib213)\], Karniadakis and Sherwin \[[2005](https://arxiv.org/html/2605.11117#bib.bib270)\], Basdevantet al\.\[[1986](https://arxiv.org/html/2605.11117#bib.bib5)\], Canutoet al\.\[[2006](https://arxiv.org/html/2605.11117#bib.bib33)\], Boyd \[[2001](https://arxiv.org/html/2605.11117#bib.bib34)\]; conservation laws with shocks activate finite\-volume and discontinuous\-Galerkin branches together with flux\-function, limiter, positivity\-preservation, and time\-integration constraintsCockburn and Shu \[[1998](https://arxiv.org/html/2605.11117#bib.bib214)\], Cockburnet al\.\[[2012](https://arxiv.org/html/2605.11117#bib.bib215)\], LeVeque \[[2002](https://arxiv.org/html/2605.11117#bib.bib218)\], Ranochaet al\.\[[2021](https://arxiv.org/html/2605.11117#bib.bib220)\], Kopriva \[[2009](https://arxiv.org/html/2605.11117#bib.bib225)\], Hesthaven and Warburton \[[2008](https://arxiv.org/html/2605.11117#bib.bib36)\], Toro \[[2013](https://arxiv.org/html/2605.11117#bib.bib32)\], Zhang and Shu \[[2003](https://arxiv.org/html/2605.11117#bib.bib3)\], Wanner and Hairer \[[1996](https://arxiv.org/html/2605.11117#bib.bib2)\]; nonlinear parameterised solves activate continuation or homotopy leaves that schedule a sequence of related problems rather than a single hard one\. These categories supply the compatibility rules GRAFT uses both when ingesting the Trixi\.jl and Nektar\+\+ documentation and when validating agent\-proposed methods against the foundational numerical\-analysis literature underlying that documentation and against the PDE class observed in the formalised problemCantwellet al\.\[[2015](https://arxiv.org/html/2605.11117#bib.bib14)\], Canutoet al\.\[[2006](https://arxiv.org/html/2605.11117#bib.bib33)\], Hesthaven and Warburton \[[2008](https://arxiv.org/html/2605.11117#bib.bib36)\], Toro \[[2013](https://arxiv.org/html/2605.11117#bib.bib32)\], LeVeque \[[2002](https://arxiv.org/html/2605.11117#bib.bib218)\]\.

These scaffolds are not claimed as new theory; their role is to initialise a scientifically meaningful action vocabulary and rule set, and the contribution of GRAFT\-ATHENA is that this vocabulary becomes a factored, inspectable, expandable, and reward\-calibrated substrate on which agents can search, revise, and accumulate experience\.

### 4\.4Error Computation

For all PIML benchmarks, the reward attached to a\(p,m\)\(p,m\)pair is the relativeL2L^\{2\}error between the predicted field and the corresponding reference solution, evaluated on a dense grid spanning the full solution domain,

R​L2=‖u^​\(x\)−u​\(x\)‖2‖u^​\(x\)‖2=∑i=1n\(u^​\(𝐱i\)−u​\(𝐱i,θ\)\)2∑i=1nu^​\(𝐱i\)2\.RL\_\{2\}=\\frac\{\\\|\\hat\{u\}\(x\)\-u\(x\)\\\|\_\{2\}\}\{\\\|\\hat\{u\}\(x\)\\\|\_\{2\}\}=\\frac\{\\sqrt\{\\sum\_\{i=1\}^\{n\}\\left\(\\hat\{u\}\(\\mathbf\{x\}\_\{i\}\)\-u\(\\mathbf\{x\}\_\{i\},\\theta\)\\right\)^\{2\}\}\}\{\\sqrt\{\\sum\_\{i=1\}^\{n\}\\hat\{u\}\(\\mathbf\{x\}\_\{i\}\)^\{2\}\}\}\.\(18\)Here,u^​\(𝐱i\)\\hat\{u\}\(\\mathbf\{x\}\_\{i\}\)denotes the reference solution at point𝐱i\\mathbf\{x\}\_\{i\}andu​\(𝐱i,θ\)u\(\\mathbf\{x\}\_\{i\},\\theta\)denotes the model prediction\. For Allen–Cahn and viscous Burgers we use the publicly available reference data released with the original studies\[McClenny and Braga\-Neto,[2023](https://arxiv.org/html/2605.11117#bib.bib203)\]; for Helmholtz, Poisson, KdV, and inviscid Burgers we use analytical solutions to computeu^\\hat\{u\}on the evaluation grid\. Unless otherwise stated, the evaluation set is a fixed dense grid ofn≈2×105n\\approx 2\\times 10^\{5\}points covering the full spatial domain \(and, for time\-dependent problems, the full space–time domain\), and is kept identical across methods to enable direct comparison; this definition is standard in the PINN/PIML literature for reporting solution\-field accuracy\[Urbánet al\.,[2024](https://arxiv.org/html/2605.11117#bib.bib496), Kiyaniet al\.,[2025](https://arxiv.org/html/2605.11117#bib.bib38), Wanget al\.,[2025a](https://arxiv.org/html/2605.11117#bib.bib44)\]\. The sameR​L2RL\_\{2\}is the observable scored by the advisor and consumed by the reward\-weighted blend of Eq\.[15](https://arxiv.org/html/2605.11117#S4.E15), so the scaffolds above and the metric here together fix both the actions in the PIML branch of𝒯A\\mathcal\{T\}\_\{A\}and the signal on which they are evaluated\.

## Acknowledgements

We acknowledge the support of the NIH grant R01AT012312, R01HL154150, MURI/AFOSR FA9550\-20\-1\-0358 project, the DOE\-MMICS SEA\-CROGS DE\-SC0023191 award, and the ONR Vannevar Bush Faculty Fellowship \(N00014\-22\-1\-2795\)\. Finally, we thank Daniel T\. Chen from Brown University for engaging in insightful discussions on tree\-structured policies and Markov decision processes\.

## Data availability

To support reproducibility, the source code for our implementation and data will be publicly available in our GitHub repository upon acceptance of the manuscript\.

##### Competing interests

The authors declare no competing interests\.

##### Author contribution

1. 1\.Conceptualization:J\.D\.T\., G\.E\.K\.
2. 2\.Methodology:J\.D\.T\.
3. 3\.Software:J\.D\.T\., Z\.C\.
4. 4\.Formal analysis:J\.D\.T\.
5. 5\.Investigation:J\.D\.T\., Z\.C\.
6. 6\.Resources:G\.E\.K\.
7. 7\.Writing – original draft:J\.D\.T\.,Z\.C\., G\.E\.K\.
8. 8\.Writing – review & editing:J\.D\.T\.,Z\.C\., G\.E\.K\.
9. 9\.Visualization:J\.D\.T\., Z\.C\.
10. 10\.Supervision:G\.E\.K\.
11. 11\.Project administration:G\.E\.K\.
12. 12\.Funding acquisition:G\.E\.K\.

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## Appendix AFormalization Details

### A\.1II\-map certification of the decision\-level factorisation

This appendix recasts the decision\-level factorisation of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)in Bayesian\-network vocabulary and certifies it as anII\-map of the joint distribution over methods, formalising the claim made informally at the end of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)that the decomposition preserves the encoded conditional\-independence assertions used by the factorisation\. Throughout, the problemppand the repository𝒟\\mathcal\{D\}are held fixed: the rowsM\(j\)​\(u,⋅\)M^\{\(j\)\}\(u,\\cdot\)of §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)are compiled, and Proposition[4\.1](https://arxiv.org/html/2605.11117#S4.Thmtheorem1)is a statement about the resulting conditional distributionπ\(⋅∣p,𝒟\)\\pi\(\\cdot\\mid p,\\mathcal\{D\}\), not the marginal over a varyingpp\. Proposition[4\.4](https://arxiv.org/html/2605.11117#S4.Thmtheorem4)is in addition a per\-step statement at iterationnn, with the within\-trial history𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}held fixed alongsideppand𝒟\\mathcal\{D\}, so the I\-map applies toPagent\(⋅∣p,𝒟,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\)P\_\{\\text\{agent\}\}\(\\cdot\\mid p,\\mathcal\{D\},\\mathsf\{History\}\_\{n\}\)for eachnn\.

The variable set is the chain set𝒞=\{C1,…,C\|𝒞\|\}\\mathcal\{C\}=\\\{C\_\{1\},\\dots,C\_\{\|\\mathcal\{C\}\|\}\\\}of §[4\.1\.3](https://arxiv.org/html/2605.11117#S4.SS1.SSS3), with chainjjrealised asaj∈𝒜j∪\{∅j\}a\_\{j\}\\in\\mathcal\{A\}\_\{j\}\\cup\\\{\\varnothing\_\{j\}\\\}\. The chain dependency graphH=\(𝒞,E∪Eρ\)H=\(\\mathcal\{C\},\\,E\\cup E\_\{\\rho\}\)of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)carries two edge families: the rule\-induced edgesEE, where each ruleRℓ∈ℛR\_\{\\ell\}\\in\\mathcal\{R\}contributes for every valuet∈Tℓtrigt\\in T^\{\\mathrm\{trig\}\}\_\{\\ell\}a directed edge fromν​\(t\)\\nu\(t\)to the affected chainj​\(ℓ\)j\(\\ell\), and the structural\-nesting edgesEρ=\{ρ​\(C\)→C:C∈𝒞,ρ​\(C\)≠⊥\}E\_\{\\rho\}=\\\{\\rho\(C\)\\to C:C\\in\\mathcal\{C\},\\ \\rho\(C\)\\neq\\bot\\\}, which encode that a subchain rooted below acc\-node is unreachable until thatcc\-parent’sss\-decision is made\. Its combined parent set is

paH​\(j\)=\{i:i→j∈E∪Eρ\},\\mathrm\{pa\}\_\{H\}\(j\)\\;=\\;\\\{\\,i\\,:\\,i\\to j\\in E\\cup E\_\{\\rho\}\\,\\\},\(19\)and we writeapaH​\(j\)=\(ai:i∈paH\(j\)\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}=\(a\_\{i\}:i\\in\\mathrm\{pa\}\_\{H\}\(j\)\)\. The level assignment of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)is the longest\-path length tojjinHH, with level0for chains with no parent inHH\. Acyclicity ofHHis a precondition: it is decided inO​\(\|𝒞\|\+\|E∪Eρ\|\)O\(\|\\mathcal\{C\}\|\+\|E\\cup E\_\{\\rho\}\|\)time at build time by Tarjan’s strongly\-connected\-components algorithmTarjan \[[1972](https://arxiv.org/html/2605.11117#bib.bib1)\]\(Algorithm[2](https://arxiv.org/html/2605.11117#alg2)\); a non\-singleton component returns a structural cycle witness for the build\-time abort message\.

Input:action knowledge graph

𝒢A\\mathcal\{G\}\_\{A\}, dependency rule set

ℛ\\mathcal\{R\}\.

Output:chain set

𝒞\\mathcal\{C\}, level assignment

level:𝒞→ℕ\\mathrm\{level\}:\\mathcal\{C\}\\to\\mathbb\{N\}\.

1ex

𝒯←Reduce​\(𝒢A\)\\mathcal\{T\}\\leftarrow\\textsc\{Reduce\}\(\\mathcal\{G\}\_\{A\}\)
\(𝒞,ν,ρ\)←Chains​\(𝒯\)\(\\mathcal\{C\},\\nu,\\rho\)\\leftarrow\\textsc\{Chains\}\(\\mathcal\{T\}\)

E←\{ν​\(t\)→ν​\(g\):Rℓ=\(hℓ,Tℓtrig,Tℓtgt,eℓ\)∈ℛ,\(t,g\)∈Tℓtrig×Tℓtgt,ν​\(t\)≠ν​\(g\)\}E\\leftarrow\\bigl\\\{\\,\\nu\(t\)\\to\\nu\(g\)\\,:\\,R\_\{\\ell\}=\(h\_\{\\ell\},T^\{\\mathrm\{trig\}\}\_\{\\ell\},T^\{\\mathrm\{tgt\}\}\_\{\\ell\},e\_\{\\ell\}\)\\in\\mathcal\{R\},\\ \(t,g\)\\in T^\{\\mathrm\{trig\}\}\_\{\\ell\}\\times T^\{\\mathrm\{tgt\}\}\_\{\\ell\},\\ \\nu\(t\)\\neq\\nu\(g\)\\,\\bigr\\\}

Eρ←\{ρ​\(C\)→C:C∈𝒞,ρ​\(C\)≠⊥\}E\_\{\\rho\}\\leftarrow\\\{\\,\\rho\(C\)\\to C\\,:\\,C\\in\\mathcal\{C\},\\ \\rho\(C\)\\neq\\bot\\,\\\}

H←\(𝒞,E∪Eρ\)H\\leftarrow\(\\mathcal\{C\},\\ E\\cup E\_\{\\rho\}\)

1ex

𝒮←Tarjan\-SCC​\(H\)\\mathcal\{S\}\\leftarrow\\textsc\{Tarjan\-SCC\}\(H\)
if*∃S∈𝒮:\|S\|\>1\\exists\\,S\\in\\mathcal\{S\}:\\ \|S\|\>1*thenabort\(“cycle witness

SSin

HH”\)

1ex

level​\(C\)←0∀C∈𝒞\\mathrm\{level\}\(C\)\\leftarrow 0\\quad\\forall\\,C\\in\\mathcal\{C\}
repeat

foreach*C∈𝒞C\\in\\mathcal\{C\}withρ​\(C\)≠⊥\\rho\(C\)\\neq\\bot*do

level​\(C\)←max⁡\(level​\(C\),level​\(ρ​\(C\)\)\+1\)\\mathrm\{level\}\(C\)\\leftarrow\\max\\bigl\(\\mathrm\{level\}\(C\),\\ \\mathrm\{level\}\(\\rho\(C\)\)\+1\\bigr\)
foreach*a→b∈Ea\\to b\\in E*do

level​\(b\)←max⁡\(level​\(b\),level​\(a\)\+1\)\\mathrm\{level\}\(b\)\\leftarrow\\max\\bigl\(\\mathrm\{level\}\(b\),\\ \\mathrm\{level\}\(a\)\+1\\bigr\)
until*no change*

return

\(𝒞,level\)\(\\mathcal\{C\},\\mathrm\{level\}\)

Algorithm 2GRAFT build\-time pipeline: action knowledge graph to factored decision levels\.###### Definition A\.1\(II\-map\)\.

A DAGHHover a finite variable setVVis anII\-map of a probability distributionPPoverVVif every conditional independence relation read offHHvia d\-separation holds inPP\.

###### Theorem A\.2\(Verma & Pearl, 1988Verma and Pearl \[[1988](https://arxiv.org/html/2605.11117#bib.bib782)\]\)\.

LetMMbe a semi\-graphoid over a finite variable setVV,θ\\thetaa total order onVV, andLθ=\(θ,B\)L\_\{\\theta\}=\(\\theta,B\)a stratified protocol ofMM, that is, for eachx∈Vx\\in V,B​\(x\)⊆predθ​\(x\)B\(x\)\\subseteq\\mathrm\{pred\}\_\{\\theta\}\(x\)andI​\(x,B​\(x\),predθ​\(x\)∖B​\(x\)\)I\\\!\\bigl\(x,\\,B\(x\),\\,\\mathrm\{pred\}\_\{\\theta\}\(x\)\\setminus B\(x\)\\bigr\)holds inMM\. Then the DAG with parent functionBBis anII\-map ofMM\.

###### Proof of Proposition[4\.1](https://arxiv.org/html/2605.11117#S4.Thmtheorem1)\.

We verify the hypotheses of Theorem[A\.2](https://arxiv.org/html/2605.11117#A1.Thmtheorem2)for the variable setV=𝒞V=\\mathcal\{C\}, total orderθ\\thetagiven by any topological extension of the level partial order of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4), and parent functionB​\(j\)=paH​\(j\)B\(j\)=\\mathrm\{pa\}\_\{H\}\(j\)\.

Acyclicity\.Tarjan’s algorithm applied toHHcertifies, in timeO​\(\|𝒞\|\+\|E∪Eρ\|\)O\(\|\\mathcal\{C\}\|\+\|E\\cup E\_\{\\rho\}\|\), that every strongly connected component is a singleton, henceHHis a DAG and the level partial order extends to a total topological orderθ\\theta\. The hypotheses of Tarjan’s theorem are met by construction:𝒞\\mathcal\{C\},EE, andEρE\_\{\\rho\}are finite;HHis directed; the rule\-expansion filter that drops triggers and targets resolving to the same chain rules outEE\-self\-loops; andρ​\(C\)≠C\\rho\(C\)\\neq Cfor everyCCrules outEρE\_\{\\rho\}\-self\-loops\.

Tail boundary\.By the construction ofπ~j\\widetilde\{\\pi\}\_\{j\}in Eq\. \([4](https://arxiv.org/html/2605.11117#S4.E4)\),π~j\\widetilde\{\\pi\}\_\{j\}is obtained fromπj\\pi\_\{j\}by composing operatorseℓ1,…,eℓre\_\{\\ell\_\{1\}\},\\dots,e\_\{\\ell\_\{r\}\}for those rules withj​\(ℓ\)=jj\(\\ell\)=jwhose triggersTℓtrigT^\{\\mathrm\{trig\}\}\_\{\\ell\}are matched by the lower\-level values\. Each triggerTℓtrigT^\{\\mathrm\{trig\}\}\_\{\\ell\}references chains inpaH​\(j\)\\mathrm\{pa\}\_\{H\}\(j\)by definition ofEE, and the structural\-nesting parent \(when present\) is inpaH​\(j\)\\mathrm\{pa\}\_\{H\}\(j\)by definition ofEρE\_\{\\rho\}, soπ~j\\widetilde\{\\pi\}\_\{j\}depends ona<level​\(j\)a\_\{<\\mathrm\{level\}\(j\)\}only throughapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}\. The activity indicatoractj\\mathrm\{act\}\_\{j\}used in Eq\. \([5](https://arxiv.org/html/2605.11117#S4.E5)\) is a function of the structural\-nesting parent’s pick and anyℱ/𝒵\\mathcal\{F\}/\\mathcal\{Z\}effects fired by triggers inpaH​\(j\)\\mathrm\{pa\}\_\{H\}\(j\), so it too depends ona<level​\(j\)a\_\{<\\mathrm\{level\}\(j\)\}only throughapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}\. Both branches ofπ¯j\\bar\{\\pi\}\_\{j\}therefore depend on the lower\-level history only throughapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}:

π¯j\(⋅\|a<level​\(j\)\)=π¯j\(⋅\|apaH​\(j\)\)\.\\bar\{\\pi\}\_\{j\}\\bigl\(\\,\\cdot\\,\\bigm\|\\,a\_\{<\\mathrm\{level\}\(j\)\}\\bigr\)\\;=\\;\\bar\{\\pi\}\_\{j\}\\bigl\(\\,\\cdot\\,\\bigm\|\\,a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}\\bigr\)\.By the chain rule alongθ\\theta, the joint factorises into the per\-chain kernelsπ¯j\(⋅∣a<level​\(j\)\)\\bar\{\\pi\}\_\{j\}\(\\,\\cdot\\,\\mid a\_\{<\\mathrm\{level\}\(j\)\}\), so the kernel equality lifts to the conditional independenceaj⟂predθ​\(j\)∖paH​\(j\)∣paH​\(j\)a\_\{j\}\\perp\\mathrm\{pred\}\_\{\\theta\}\(j\)\\setminus\\mathrm\{pa\}\_\{H\}\(j\)\\mid\\mathrm\{pa\}\_\{H\}\(j\)in the policy\-induced distribution, which is the tail\-boundary condition forB​\(j\)=paH​\(j\)B\(j\)=\\mathrm\{pa\}\_\{H\}\(j\)\.

Conclusion\.The semi\-graphoid axioms hold for any probability distribution; the topological order is supplied by acyclicity; the tail boundary holds by the previous step\. Theorem[A\.2](https://arxiv.org/html/2605.11117#A1.Thmtheorem2)then yields thatHHwith parent functionpaH\\mathrm\{pa\}\_\{H\}is anII\-map ofπ\(⋅∣p,𝒟\)\\pi\(\\,\\cdot\\,\\mid p,\\mathcal\{D\}\), and substituting the parent\-set conditioning into Eq\. \([6](https://arxiv.org/html/2605.11117#S4.E6)\) gives Eq\. \([7](https://arxiv.org/html/2605.11117#S4.E7)\)\. ∎

TheII\-map property is what formalises the informal claim that the decomposition is structurally conservative: every conditional independence read offHHvia d\-separation holds inπ\(⋅∣p,𝒟\)\\pi\(\\,\\cdot\\,\\mid p,\\mathcal\{D\}\), so no coupling encoded inℛ\\mathcal\{R\}or in the nesting structure of𝒯\\mathcal\{T\}is silently dropped by the per\-chain factorisation\. The converse direction \(faithfulness, every independence inπ\(⋅∣p,𝒟\)\\pi\(\\,\\cdot\\,\\mid p,\\mathcal\{D\}\)has a d\-separation witness inHH\) is not claimed:HHmay be a non\-minimalII\-map, which is the standard situation for hand\-specified Bayesian networks\.

###### Proof of Proposition[4\.4](https://arxiv.org/html/2605.11117#S4.Thmtheorem4)\.

We verify the hypotheses of Theorem[A\.2](https://arxiv.org/html/2605.11117#A1.Thmtheorem2)forV=𝒞V=\\mathcal\{C\}, the same total orderθ\\thetaused in the proof of Proposition[4\.1](https://arxiv.org/html/2605.11117#S4.Thmtheorem1), and parent functionB​\(j\)=paH​\(j\)B\(j\)=\\mathrm\{pa\}\_\{H\}\(j\)\. Acyclicity and the topological order are inherited from that proof; only the tail boundary changes, since the distribution is nowPagentP\_\{\\text\{agent\}\}\.

Tail boundary\.The I\-map is a per\-step statement at iterationnn, with𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\\mathsf\{History\}\_\{n\}treated as fixed parameters of the policy at that step \(analogous to the rule operators of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)\)\. By Assumption[4\.3](https://arxiv.org/html/2605.11117#S4.Thmtheorem3), the proposer’s policy at chainjjdepends onpredθ​\(j\)\\mathrm\{pred\}\_\{\\theta\}\(j\)only throughapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\},hintsj\\mathrm\{hints\}\_\{j\}, and the per\-chain history projection𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\(j\)\\mathsf\{History\}\_\{n\}^\{\(j\)\}, so

Pagent​\(aj\|predθ​\(j\),hintsj,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\(j\)\)=Pagent​\(aj\|paH​\(j\),hintsj,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\(j\)\)\.P\_\{\\text\{agent\}\}\\\!\\bigl\(a\_\{j\}\\,\\big\|\\,\\mathrm\{pred\}\_\{\\theta\}\(j\),\\,\\mathrm\{hints\}\_\{j\},\\,\\mathsf\{History\}\_\{n\}^\{\(j\)\}\\bigr\)\\;=\\;P\_\{\\text\{agent\}\}\\\!\\bigl\(a\_\{j\}\\,\\big\|\\,\\mathrm\{pa\}\_\{H\}\(j\),\\,\\mathrm\{hints\}\_\{j\},\\,\\mathsf\{History\}\_\{n\}^\{\(j\)\}\\bigr\)\.\(20\)Sincehintsj=hj​\(j,apaH​\(j\)\)\\mathrm\{hints\}\_\{j\}=h\_\{j\}\(j,a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}\)is a deterministic function ofapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}, conditioning onhintsj\\mathrm\{hints\}\_\{j\}is redundant givenapaH​\(j\)a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\}; and since𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\(j\)\\mathsf\{History\}\_\{n\}^\{\(j\)\}is fixed at stepnn\(a parameter of the per\-step policy, not a random variable in the joint over𝒞\\mathcal\{C\}\), it can be absorbed into the policy specification without affecting the conditional\-independence statement, so

aj⟂predθ​\(j\)∖paH​\(j\)\|paH​\(j\)a\_\{j\}\\;\\perp\\;\\mathrm\{pred\}\_\{\\theta\}\(j\)\\setminus\\mathrm\{pa\}\_\{H\}\(j\)\\;\\bigm\|\\;\\mathrm\{pa\}\_\{H\}\(j\)\(21\)inPagentP\_\{\\text\{agent\}\}, which is the tail\-boundary condition forB​\(j\)=paH​\(j\)B\(j\)=\\mathrm\{pa\}\_\{H\}\(j\)\. The critic clause of Assumption[4\.3](https://arxiv.org/html/2605.11117#S4.Thmtheorem3)ensures that acceptance is a function only of\(aj,apaH​\(j\),M\(j\)​\(u,⋅\),hintsj\)\(a\_\{j\},a\_\{\\mathrm\{pa\}\_\{H\}\(j\)\},M^\{\(j\)\}\(u,\\cdot\),\\mathrm\{hints\}\_\{j\}\), so the rejection\-resampling step at chainjjreweightsaja\_\{j\}within the parent context without introducing dependence on non\-parent predecessors, preserving the conditional structure of Eq\. \([20](https://arxiv.org/html/2605.11117#A1.E20)\)\.

Conclusion\.Theorem[A\.2](https://arxiv.org/html/2605.11117#A1.Thmtheorem2)yields thatHHwith parent functionpaH\\mathrm\{pa\}\_\{H\}is anII\-map ofPagent\(⋅∣p,𝒟,𝖧𝗂𝗌𝗍𝗈𝗋𝗒n\)P\_\{\\text\{agent\}\}\(\\,\\cdot\\,\\mid p,\\mathcal\{D\},\\mathsf\{History\}\_\{n\}\)for each iterationnn, and the chain rule underθ\\thetacollapses to Eq\. \([16](https://arxiv.org/html/2605.11117#S4.E16)\)\. ∎

### A\.2Partition of Unity Projection

This appendix supplies the recursive layout procedure \(Algorithm[3](https://arxiv.org/html/2605.11117#alg3)\) for the embeddingΦ\\Phiof §[4\.1\.5](https://arxiv.org/html/2605.11117#S4.SS1.SSS5), together with the proof of Proposition[4\.2](https://arxiv.org/html/2605.11117#S4.Thmtheorem2)\. The uniform\-children structural assumption the proof relies on is stated in the body\.

Input:rooted tree

𝒯\\mathcal\{T\}with root

rrand edge\-type labels

τ∈\{s,c\}\\tau\\in\\\{s,c\\\}\.

Output:

Φ:V​\(𝒯\)→\[0,1\]3\\Phi:V\(\\mathcal\{T\}\)\\to\[0,1\]^\{3\}\.

1exprocedure

Layout​\(v,\[x0,x1\],\[y0,y1\],d\)\\textsc\{Layout\}\(v,\\,\[x\_\{0\},x\_\{1\}\],\\,\[y\_\{0\},y\_\{1\}\],\\,d\)
Φ​\(v\)←\(\(x0\+x1\)/2,\(y0\+y1\)/2,d\)\\Phi\(v\)\\leftarrow\\bigl\(\(x\_\{0\}\+x\_\{1\}\)/2,\\ \(y\_\{0\}\+y\_\{1\}\)/2,\\ d\\bigr\)

if*vvhas no children*thenreturn

partition children of

vvby edge type into groups

\{Cτ\}τ\\\{C\_\{\\tau\}\\\}\_\{\\tau\}
foreach*groupCτC\_\{\\tau\}, with children sorted by name*do

n←\|Cτ\|n\\leftarrow\|C\_\{\\tau\}\|
if*nneven*then

q←nq\\leftarrow n;

k←−1k\\leftarrow\-1//no skip

else

q←n\+1q\\leftarrow n\+1;

k←⌈q/2⌉k\\leftarrow\\lceil q/2\\rceil//skip middle

U←\(0,1,…,q−1\)U\\leftarrow\(0,1,\\dots,q\-1\)with index

kkremoved

if*τ=c\\tau=c*then

h←\(y1−y0\)/qh\\leftarrow\(y\_\{1\}\-y\_\{0\}\)/q
foreach*\(i,u\)∈zip​\(U,Cτ\)\(i,u\)\\in\\mathrm\{zip\}\(U,C\_\{\\tau\}\)*do

Layout​\(u,\[x0,x1\],\[y0\+i​h,y0\+\(i\+1\)​h\],d\+1\)\\textsc\{Layout\}\\bigl\(u,\\ \[x\_\{0\},x\_\{1\}\],\\ \[y\_\{0\}\+i\\,h,\\ y\_\{0\}\+\(i\+1\)\\,h\],\\ d\+1\\bigr\)
else//

τ=s\\tau=s
w←\(x1−x0\)/qw\\leftarrow\(x\_\{1\}\-x\_\{0\}\)/q
foreach*\(i,u\)∈zip​\(U,Cτ\)\(i,u\)\\in\\mathrm\{zip\}\(U,C\_\{\\tau\}\)*do

Layout​\(u,\[x0\+i​w,x0\+\(i\+1\)​w\],\[y0,y1\],d\+1\)\\textsc\{Layout\}\\bigl\(u,\\ \[x\_\{0\}\+i\\,w,\\ x\_\{0\}\+\(i\+1\)\\,w\],\\ \[y\_\{0\},y\_\{1\}\],\\ d\+1\\bigr\)
end procedure

1ex

Layout​\(r,\[0,1\],\[0,1\],0\)\\textsc\{Layout\}\(r,\\ \[0,1\],\\ \[0,1\],\\ 0\)
D←maxv∈V​\(𝒯\)⁡d​\(v\)D\\leftarrow\\max\_\{v\\in V\(\\mathcal\{T\}\)\}d\(v\)

foreach*v∈V​\(𝒯\)v\\in V\(\\mathcal\{T\}\)*do

z​\(v\)←d​\(v\)/Dz\(v\)\\leftarrow d\(v\)/D
return

Φ\\Phi

Algorithm 3PartitionOfUnityLayout: recursive subdivision with parent\-midpoint exclusion\.For each nodev∈V​\(𝒯\)v\\in V\(\\mathcal\{T\}\), letR​\(v\)⊂\[0,1\]2R\(v\)\\subset\[0,1\]^\{2\}denote the closed rectangle owned byvvduring the recursion of Algorithm[3](https://arxiv.org/html/2605.11117#alg3):R​\(root\)=\[0,1\]2R\(\\textsc\{root\}\)=\[0,1\]^\{2\}, and for each childccofvvthe rectangleR​\(c\)R\(c\)is the slot rectangle constructed by the recursive call toLayout\. By constructionπx​y​\(Φ​\(v\)\)=centroid​\(R​\(v\)\)\\pi\_\{xy\}\(\\Phi\(v\)\)=\\mathrm\{centroid\}\(R\(v\)\)\.

###### Proof of Proposition[4\.2](https://arxiv.org/html/2605.11117#S4.Thmtheorem2)\.

We establish three sub\-claims and then dispatch by lowest common ancestor\.

\(C1\) Containment\.For every descendantuuofvv,R​\(u\)⊆R​\(v\)R\(u\)\\subseteq R\(v\), andR​\(u\)R\(u\)is a non\-degenerate \(positive\-measure\) closed rectangle\. By induction on path length: each recursive step constructsR​\(c\)R\(c\)as a slot ofR​\(v\)R\(v\)along the active axis with width\(b−a\)/q\>0\(b\-a\)/q\>0forq≥1q\\geq 1, so containment and non\-degeneracy are preserved\.

\(C2\) Sibling rectangles have disjoint interiors\.Under the uniform\-children assumption, all children ofvvare processed in a single group with a single active axis\. Their slot indicesi,i′∈Ui,i^\{\\prime\}\\in Uare distinct integers, giving disjoint open slot intervals along the active axis\. The inactive axis is shared, so the rectangles tilevv’s active\-axis interval without overlap; their interiors are pairwise disjoint\.

\(C3\) Centroid interior to own rectangle, but not to any child’s rectangle\.The centroid of a non\-degenerate axis\-aligned rectangle lies in its topological interior; combined with \(C1\),centroid​\(R​\(v\)\)∈int​\(R​\(v\)\)\\mathrm\{centroid\}\(R\(v\)\)\\in\\mathrm\{int\}\(R\(v\)\)\. We show further thatcentroid​\(R​\(v\)\)∉int​\(R​\(c\)\)\\mathrm\{centroid\}\(R\(v\)\)\\notin\\mathrm\{int\}\(R\(c\)\)for any childccofvv\. NormalizeR​\(v\)R\(v\)’s active\-axis interval to\[0,1\]\[0,1\], socentroid​\(R​\(v\)\)\\mathrm\{centroid\}\(R\(v\)\)has active\-axis coordinate1/21/2\.

*Evennn:*q=nq=nis even, slotiihas active\-axis interval\[i/q,\(i\+1\)/q\]\[i/q,\(i\+1\)/q\], and1/2=\(q/2\)/q1/2=\(q/2\)/qis the boundary between slotsq/2−1q/2\-1andq/2q/2\. Both slots are populated, so1/21/2is the shared boundary ofR​\(cq/2−1\)R\(c\_\{q/2\-1\}\)andR​\(cq/2\)R\(c\_\{q/2\}\), in the interior of neither\.

*Oddnn:*q=n\+1q=n\+1is even and the skipped slot isk=q/2k=q/2, with active\-axis interval\[1/2,1/2\+1/q\]\[1/2,1/2\+1/q\]\. The point1/21/2is the right boundary of slotq/2−1q/2\-1\(populated\) and the left boundary of slotq/2q/2\(empty\)\. Hence1/21/2lies on the boundary ofR​\(cq/2−1\)R\(c\_\{q/2\-1\}\)and outside every other child’s rectangle, in the interior of none\.

In both casescentroid​\(R​\(v\)\)\\mathrm\{centroid\}\(R\(v\)\)is on the boundary of at most one child’s rectangle and not in any child’s interior\. Since for any descendantuuof any childcc,centroid​\(R​\(u\)\)∈int​\(R​\(u\)\)⊆int​\(R​\(c\)\)\\mathrm\{centroid\}\(R\(u\)\)\\in\\mathrm\{int\}\(R\(u\)\)\\subseteq\\mathrm\{int\}\(R\(c\)\)\(the latter inclusion is standard for axis\-aligned sub\-rectangles\), no descendant centroid can equalcentroid​\(R​\(v\)\)\\mathrm\{centroid\}\(R\(v\)\)\.

Case analysis\.Suppose, for contradiction, thatu≠vu\\neq vwithπx​y​\(Φ​\(u\)\)=πx​y​\(Φ​\(v\)\)\\pi\_\{xy\}\(\\Phi\(u\)\)=\\pi\_\{xy\}\(\\Phi\(v\)\)\. Letwwbe the lowest common ancestor ofuuandvv\.

*Case A:w∈\{u,v\}w\\in\\\{u,v\\\}\.*Without loss of generalityw=vw=v, souuis a strict descendant ofvv\. By \(C3\),centroid​\(R​\(u\)\)∈int​\(R​\(c\)\)\\mathrm\{centroid\}\(R\(u\)\)\\in\\mathrm\{int\}\(R\(c\)\)for some childccofvv, whilecentroid​\(R​\(v\)\)\\mathrm\{centroid\}\(R\(v\)\)is not inint​\(R​\(c\)\)\\mathrm\{int\}\(R\(c\)\)\. Contradiction\.

*Case B:u,vu,vare in disjoint subtrees rooted at distinct childrencu,cvc\_\{u\},c\_\{v\}ofww\.*By \(C2\),R​\(cu\)R\(c\_\{u\}\)andR​\(cv\)R\(c\_\{v\}\)have disjoint interiors\. By \(C1\) and the standard inclusion for axis\-aligned sub\-rectangles,centroid​\(R​\(u\)\)∈int​\(R​\(cu\)\)\\mathrm\{centroid\}\(R\(u\)\)\\in\\mathrm\{int\}\(R\(c\_\{u\}\)\)andcentroid​\(R​\(v\)\)∈int​\(R​\(cv\)\)\\mathrm\{centroid\}\(R\(v\)\)\\in\\mathrm\{int\}\(R\(c\_\{v\}\)\)\. Disjoint open sets cannot share a point, so the two centroids differ\. Contradiction\.

Both cases yield a contradiction, henceπx​y∘Φ\\pi\_\{xy\}\\circ\\Phiis injective\. The full embeddingΦ\\Phiinherits injectivity since it carries strictly more information thanπx​y∘Φ\\pi\_\{xy\}\\circ\\Phi\. ∎

*Discretization for fingerprinting\.*The fingerprint of Eq\.[10](https://arxiv.org/html/2605.11117#S4.E10)discretizes the continuous embedding by binning\(x,y\)\(x,y\)at resolutionKKand pairing with the integer depthdd, using the boundary\-clamped bin mapβK\\beta\_\{K\}defined in Eq\.[9](https://arxiv.org/html/2605.11117#S4.E9)\. By Proposition[4\.2](https://arxiv.org/html/2605.11117#S4.Thmtheorem2), the continuous coordinates are pairwise distinct on the finite setV​\(𝒯\)V\(\\mathcal\{T\}\), so there exists at least one finiteKKat whichβK\\beta\_\{K\}is injective onV​\(𝒯\)V\(\\mathcal\{T\}\)\. We do not claim injectivity for everyK≥K⋆K\\geq K^\{\\star\}: floor binning shifts grid lines withKK, so injectivity at one resolution does not imply injectivity at every larger one\. The role of Algorithm[4](https://arxiv.org/html/2605.11117#alg4)is therefore not to*secure*identity for an interval of resolutions but to choose a single operating resolutionK⋆K^\{\\star\}at whichβK\\beta\_\{K\}is injective on the currentV​\(𝒯\)V\(\\mathcal\{T\}\)and at which the per\-cell footprint is small enough to produce a usable visualization on a finite grid\. We pickK⋆K^\{\\star\}as the smallest suchKK, with a hard capKmax=4096K\_\{\\max\}=4096as a numerical safety stop\. All downstream uses of fingerprints \(neighbour ranking, policy aggregation\) are evaluated atK=K⋆K=K^\{\\star\}\.

Input:embedding

Φ\\Phi, depth map

dd, cap

KmaxK\_\{\\max\}\(implementation:

Kmax=4096K\_\{\\max\}=4096\)\.

Output:

K⋆K^\{\\star\}, the minimum resolution at which

βK\\beta\_\{K\}is injective on

V​\(𝒯\)V\(\\mathcal\{T\}\)\.

1exfor*K←1K\\leftarrow 1toKmaxK\_\{\\max\}*do

cells←∅\\textit\{cells\}\\leftarrow\\emptyset;

ok←𝐭𝐫𝐮𝐞\\textit\{ok\}\\leftarrow\\mathbf\{true\}
foreach*v∈V​\(𝒯\)v\\in V\(\\mathcal\{T\}\)*do

if*βK​\(v\)∈cells\\beta\_\{K\}\(v\)\\in\\textit\{cells\}*then

ok←𝐟𝐚𝐥𝐬𝐞\\textit\{ok\}\\leftarrow\\mathbf\{false\};break

else

cells←cells∪\{βK​\(v\)\}\\textit\{cells\}\\leftarrow\\textit\{cells\}\\cup\\\{\\beta\_\{K\}\(v\)\\\}
if*ok*thenreturn

KK
abort\(“no

K≤KmaxK\\leq K\_\{\\max\}separates all nodes”\)

Algorithm 4MinKForUniqueCells— smallest grid resolution at which the binned cells ofV​\(𝒯\)V\(\\mathcal\{T\}\)remain pairwise distinct\.The capKmax=4096K\_\{\\max\}=4096is a numerical safety stop, not a correctness condition: by Proposition[4\.2](https://arxiv.org/html/2605.11117#S4.Thmtheorem2)the continuous coordinates ofV​\(𝒯\)V\(\\mathcal\{T\}\)are pairwise distinct, so some finiteKKat whichβK\\beta\_\{K\}is injective exists\. The cap bounds the implementation’s search for the smallest suchKK; if it is hit before injectivity is achieved on the currentV​\(𝒯\)V\(\\mathcal\{T\}\), the tree’s active\-axis gaps have shrunk past the chosen grid budget — a visualization concern \(the chosenKmaxK\_\{\\max\}is too small for identity\-preserving rendering on this tree\), not a failure of the continuous embedding\. OnceK⋆K^\{\\star\}is fixed, the fingerprint of a problemPiP\_\{i\}in Eq\.[10](https://arxiv.org/html/2605.11117#S4.E10)reduces toF​\(Pi,K⋆\)=\{βK⋆​\(v\):v∈Pi∩Vkeep\}F\(P\_\{i\},K^\{\\star\}\)=\\\{\\beta\_\{K^\{\\star\}\}\(v\):v\\in P\_\{i\}\\cap V\_\{\\mathrm\{keep\}\}\\\}, and this fingerprint set is the problem’s identifier under the current tree version and the chosenK⋆K^\{\\star\}\(no second\-stage hashing is needed\)\.

### A\.3Fingerprint distance is a metric

The neighbour ranking and policy aggregation of §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6)use the fingerprint JaccardJKJ\_\{K\}of Eq\. \([11](https://arxiv.org/html/2605.11117#S4.E11)\) as a similarity\. The complementary form1−JK1\-J\_\{K\}is in fact a true metric on the space of non\-empty fingerprint sets, which is what makes distance\-based notions on the substrate \(nearest neighbours, balls, triangle inequality\) well\-defined\.

###### Proposition A\.3\(Fingerprint distance is a metric\)\.

For non\-empty finite setsA,B⊆ℤ3A,B\\subseteq\\mathbb\{Z\}^\{3\}, define

dJ​\(A,B\)=1−\|A∩B\|\|A∪B\|\.d\_\{J\}\(A,B\)=1\-\\frac\{\|A\\cap B\|\}\{\|A\\cup B\|\}\.\(22\)ThendJd\_\{J\}is a metric on the familyℱ\\mathcal\{F\}of non\-empty finite subsets ofℤ3\\mathbb\{Z\}^\{3\}\. Applied separately on the problem tree𝒯P\\mathcal\{T\}\_\{P\}and the method tree𝒯A\\mathcal\{T\}\_\{A\}, and assuming for every realised object the kept\-node encodingPi↦Pi∩VkeepP\_\{i\}\\mapsto P\_\{i\}\\cap V\_\{\\text\{keep\}\}is non\-empty and injective,dJd\_\{J\}is a metric on the realised problem fingerprints in𝒟\\mathcal\{D\}, on the realised method fingerprints in𝒟\\mathcal\{D\}, and pulls back to a metric on the realised problems and on the realised methods\. The cell\-level injectivity needed for the pullback comes fromβK⋆\\beta\_\{K^\{\\star\}\}being injective onV​\(𝒯\)V\(\\mathcal\{T\}\)\(Algorithm[4](https://arxiv.org/html/2605.11117#alg4)\); the object\-level injectivity of the kept\-node encoding is an additional condition, and it holds under the defaultVkeepV\_\{\\text\{keep\}\}ofss\-leaves since two problems making identicalss\-decisions on every chain are operationally identical\.

###### Proof\.

Non\-negativity follows from0≤\|A∩B\|≤\|A∪B\|0\\leq\|A\\cap B\|\\leq\|A\\cup B\|, anddJ​\(A,A\)=0d\_\{J\}\(A,A\)=0by inspection\. Identity holds sincedJ​\(A,B\)=0d\_\{J\}\(A,B\)=0iff\|A∩B\|=\|A∪B\|\|A\\cap B\|=\|A\\cup B\|iffA=BA=B\. Symmetry follows from\|A∩B\|=\|B∩A\|\|A\\cap B\|=\|B\\cap A\|and\|A∪B\|=\|B∪A\|\|A\\cup B\|=\|B\\cup A\|\. The triangle inequality is the non\-trivial axiom: rewritingdJ​\(A,B\)=\|A​△​B\|/\|A∪B\|d\_\{J\}\(A,B\)=\|A\\triangle B\|/\|A\\cup B\|and applying the symmetric\-difference triangle inequality\|A​△​C\|≤\|A​△​B\|\+\|B​△​C\|\|A\\triangle C\|\\leq\|A\\triangle B\|\+\|B\\triangle C\|together with the Steinhaus normalisation argument yieldsdJ​\(A,C\)≤dJ​\(A,B\)\+dJ​\(B,C\)d\_\{J\}\(A,C\)\\leq d\_\{J\}\(A,B\)\+d\_\{J\}\(B,C\)\. The detailed chain is given inLipkus \[[1999](https://arxiv.org/html/2605.11117#bib.bib784)\]; an earlier set\-based statement appears inLevandowsky and Winter \[[1971](https://arxiv.org/html/2605.11117#bib.bib13)\]\. ∎

Two consequences for the substrate, on each tree separately: balls

\{P:dJ​\(F​\(P,K⋆\),F​\(pnew,K⋆\)\)≤r\}\\\{P:d\_\{J\}\(F\(P,K^\{\\star\}\),F\(p\_\{\\text\{new\}\},K^\{\\star\}\)\)\\leq r\\\}around an incoming problempnewp\_\{\\text\{new\}\}are well\-defined neighbourhoods on the problem\-fingerprint subspace of𝒟\\mathcal\{D\}, with an analogous construction on the method\-fingerprint subspace, and the triangle inequality bounds two\-hop closeness in each\. The neighbour selection byJK⋆J\_\{K^\{\\star\}\}in Eq\. \([12](https://arxiv.org/html/2605.11117#S4.E12)\) therefore corresponds to ranking by a true distance, not merely by an unstructured similarity\.

### A\.4Policy update procedure

Algorithm[5](https://arxiv.org/html/2605.11117#alg5)is the pseudocode bundling of the policy and prior update of §[4\.1\.6](https://arxiv.org/html/2605.11117#S4.SS1.SSS6): it ranks𝒟\\mathcal\{D\}by the fingerprint JaccardJK⋆J\_\{K^\{\\star\}\}, builds per\-neighbour partial specificationsSiS\_\{i\}, blends them row\-wise intoMdataM\_\{\\text\{data\}\}and then with the uniform priorμ\\muto obtainMM, and finally applies the rule operators of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)\. Every step is one of Eqs\. \([12](https://arxiv.org/html/2605.11117#S4.E12)\)–\([15](https://arxiv.org/html/2605.11117#S4.E15)\); the algorithm fixes the order in which they are evaluated\.

Input:new problem

pnewp\_\{\\text\{new\}\}; dataset

𝒟\\mathcal\{D\}; neighbour count

NnbrN\_\{\\text\{nbr\}\}; uniform prior

μ\\mu; rule set

ℛ\\mathcal\{R\}\.

Output:updated rows

M\(j\)​\(u,⋅\)M^\{\(j\)\}\(u,\\cdot\)for every chain

jjand every internal

u∈𝒯Au\\in\\mathcal\{T\}\_\{A\}\.

1exrank

𝒟\\mathcal\{D\}by

sim​\(pnew,⋅\):=JK⋆​\(pnew,⋅\)\\mathrm\{sim\}\(p\_\{\\text\{new\}\},\\cdot\):=J\_\{K^\{\\star\}\}\(p\_\{\\text\{new\}\},\\cdot\)on

ss\-leaves; keep top

NnbrN\_\{\\text\{nbr\}\}
foreach*neighbouriiin the topN*nbr*N\_\{\\text\{nbr\}\}*do

compute the weight

wiw\_\{i\}from Eq\.[12](https://arxiv.org/html/2605.11117#S4.E12)

build

SiS\_\{i\}by Eq\.[13](https://arxiv.org/html/2605.11117#S4.E13)\(one\-hot on path,

μ\\muoff path\)

foreach*internal nodeu∈𝒯Au\\in\\mathcal\{T\}\_\{A\}*do

set

Mdata​\(u,⋅\)M\_\{\\text\{data\}\}\(u,\\cdot\)by Eq\.[14](https://arxiv.org/html/2605.11117#S4.E14)

set

M​\(u,⋅\)←W¯⋅Mdata​\(u,⋅\)\+\(1−W¯\)⋅μ​\(u,⋅\)M\(u,\\cdot\)\\leftarrow\\bar\{W\}\\cdot M\_\{\\text\{data\}\}\(u,\\cdot\)\+\(1\-\\bar\{W\}\)\\cdot\\mu\(u,\\cdot\)via Eq\.[15](https://arxiv.org/html/2605.11117#S4.E15)

foreach*ruleRℓ∈ℛR\_\{\\ell\}\\in\\mathcal\{R\}firing onuu*do

update

M​\(u,⋅\)M\(u,\\cdot\)via the operator

eℓ∈\{ℱ,𝒵\}e\_\{\\ell\}\\in\\\{\\mathcal\{F\},\\mathcal\{Z\}\\\}of §[4\.1\.4](https://arxiv.org/html/2605.11117#S4.SS1.SSS4)

return

MM

Algorithm 5GRAFT policy and prior update at problem arrival\.

## Appendix BGRAFT\-ATHENA Run Records

### B\.1Apollo command module at Mach 10

This appendix records the full agent trajectory behind the result of §[2\.5](https://arxiv.org/html/2605.11117#S2.SS5), stage by stage\. The case is the hypersonic inviscid flow over the Apollo command module forebody atMa∞=10\\mathrm\{Ma\}\_\{\\infty\}=10, with geometry taken from Griffith & Boylan’s 1968 postflight aerodynamics reportGriffith and Boylan \[[1968](https://arxiv.org/html/2605.11117#bib.bib16)\]\. The complete artefacts \(planning documents, method JSON, mesh, elixir, postprocessing scripts, diagnostic report\) are archived in the run directoryEuler\_20260419\_2357; verbatim quotations below are drawn from those documents and cited inline by file\. The run completed in a single iteration, with the Stage\-13 advisor returning APPROVED and a reward of100/100100/100with no edits prescribed\.

#### B\.1\.1Stage 1 — problem specification

The user request fixes a 2D axisymmetric compressible Euler problem in the meridional half\-plane\(z,r\)\(z,r\)withr≥0r\\geq 0, governed by

∂t𝐔\+∂z𝐅​\(𝐔\)\+∂r𝐆​\(𝐔\)\+𝐒​\(𝐔,r\)=0,𝐒=1r​\(ρ​ur,ρ​uz​ur,ρ​ur2,\(ρ​E\+p\)​ur\)⊤,\\partial\_\{t\}\\mathbf\{U\}\+\\partial\_\{z\}\\mathbf\{F\}\(\\mathbf\{U\}\)\+\\partial\_\{r\}\\mathbf\{G\}\(\\mathbf\{U\}\)\+\\mathbf\{S\}\(\\mathbf\{U\},r\)=0,\\quad\\mathbf\{S\}=\\tfrac\{1\}\{r\}\\bigl\(\\rho u\_\{r\},\\ \\rho u\_\{z\}u\_\{r\},\\ \\rho u\_\{r\}^\{2\},\\ \(\\rho E\+p\)u\_\{r\}\\bigr\)^\{\\\!\\top\},closed byp=\(γ−1\)​\(ρ​E−12​ρ​\(uz2\+ur2\)\)p=\(\\gamma\-1\)\\bigl\(\\rho E\-\\tfrac\{1\}\{2\}\\rho\(u\_\{z\}^\{2\}\+u\_\{r\}^\{2\}\)\\bigr\)withγ=1\.4\\gamma=1\.4\. Geometry follows the “Symmetrical Smooth Heat Shield” configuration ofGriffith and Boylan \[[1968](https://arxiv.org/html/2605.11117#bib.bib16)\]: nose radiusRn=4\.694​mR\_\{n\}=4\.694\\,\\mathrm\{m\}, base radiusRb=1\.956​mR\_\{b\}=1\.956\\,\\mathrm\{m\}, shoulder filletrc=0\.196​mr\_\{c\}=0\.196\\,\\mathrm\{m\}, cone half\-angleθcone=33∘\\theta\_\{\\mathrm\{cone\}\}=33^\{\\circ\}, axial lengthL≈3\.621​mL\\approx 3\.621\\,\\mathrm\{m\}\. Boundary conditions are inviscid slip on the capsule, axisymmetric symmetry onr=0r=0, supersonic Dirichlet on the upstream and outer surfaces, and supersonic extrapolation on the downstream surface\. Non\-dimensionalization usesρ∞=1\\rho\_\{\\infty\}=1,p∞=1/γp\_\{\\infty\}=1/\\gamma,a∞=1a\_\{\\infty\}=1,u∞=M∞=10u\_\{\\infty\}=M\_\{\\infty\}=10\.

#### B\.1\.2Stage 4 — method selection

Method selection traverses a nine\-level decision tree, with each level resolving a structural constraint inherited from levels above\. The level\-4 decision is the binding one: the system chooses subcell invariant\-domain\-preserving limiting \(Trixi\_SubcellLimiterIDP\) over the standard Hennemann–Gassner blending indicator\. The recorded justification is

> “Hint\[\[h​10\]\]\[\\\!\[h10\]\\\!\]explicitly recommendsTrixi\_SubcellLimiterIDPfor stagnation\-positivity problems like a detached bow shock at Mach 10, because standard HG blending is not unconditionally positivity\-preserving and will crash at the stagnation node\.” method\_refinement\_log\.json, level\_4 critic\_explanation

At level 5 the system enforces the consequence: invariant\-domain\-preserving subcell limiting blocks adaptive mesh refinement, so AMR is removed and curved boundaries are activated to avoid faceting at the shock,

> “Earlier we selectedTrixi\_SubcellLimiterIDP, and DEPd​15d15explicitly says IDP blocks the entire AMR umbrella, so the current AMR seed must be removed\. … this blunt\-body Apollo forebody has genuinely curved geometry \(spherical heatshield plus rounded shoulder fillet\), and for hypersonic bow\-shock/stagnation resolution a faceted straight\-edged wall is a poor geometric match\. So despite the seed, I switch the curvature leaf to MS: Curved\.” method\_refinement\_log\.json, level\_5 reasoning

The full leaf path through the nine levels is summarized in Table[2](https://arxiv.org/html/2605.11117#A2.T2)\.

Table 2:Final method chain selected by GRAFT\-ATHENA for the Apollo Mach\-10 case\.Levels 4 and 5 \(bold\) are the load\-bearing decisions: subcell IDP at level 4 forces no\-AMR plus curved boundaries at level 5, and the time\-integrator and CFL choices at levels 6–8 inherit the CFL\-bounded SSP\-RK constraint that IDP imposes\.
#### B\.1\.3Stages 5–6 — mesh planning and generation

The mesh\-planning record formulates the mesh as a direct consequence of the level\-4/5 method choice rather than as an independent design exercise,

> “IDP is incompatible with AMR \(hint L4h10 \+ dependency d7\): AMR nodes are off \(L5→\\toTrixi\_NoAMR,Trixi\_AMR\_NoIndicator,Trixi\_NoAMRCallback\)\. Consequence for the mesh: the base mesh must be pre\-graded at plan time and clustered around every feature the physics produces; runtime adaptation is unavailable\.” Meshing\_Details\.md, lines 63–68

The system then anticipates the failure mode the mesh has to defeat,

> “Resolve the detached bow shock atM∞=10M\_\{\\infty\}=10; this is the strong stationary shock the IDP limiter is actually selected for\. Under\-resolution there will either quench the shock across too many cells \(smearing\) or crash positivity at the stagnation node\.” Meshing\_Details\.md, lines 139–141

and articulates exactly where the IDP bounds will be binding, motivating the densest cluster,

> “Subsonic stagnation pocket near the nose\. Resolve the subsonic pocket between the bow shock and the nose, whereMlocal→0M\_\{\\mathrm\{local\}\}\\to 0,ρ\\rhoandppapproach their local extremes, and IDP’s two\-sided invariant\-domain bounds are binding\.” Meshing\_Details\.md, lines 152–156

The realized mesh is a conformal quadrilateral grid of48684868elements with3838analytically\-curved capsule edges,polydeg=4\\mathrm\{polydeg\}=4, peak aspect ratio2\.222\.22, peak skewness0\.5480\.548, and growth\-ratep95=1\.208\\mathrm\{p95\}=1\.208\(max1\.7021\.702\)\. Six refinement zones are explicitly named in the planning document: bow\-shock band, subsonic pocket, shoulder fillet, heatshield arc, wake corridor, and outer grading shell\.

#### B\.1\.4Stage 8 — implementation

The implementation translates the method chain into a Trixi\.jl elixir\. The salient decisions, all recorded ahead of any execution, are:flux\_ranocha\_turboas the entropy\-conservative DG volume flux paired withflux\_lax\_friedrichsas a dissipative surface flux; a three\-constraint IDP limiter enforcing positivity ofρ\\rho, positivity ofpp, and a one\-sided lower bound on Guermond entropy;SimpleSSPRK33time integration withStepsizeCallbackatCFL=0\.5\\mathrm\{CFL\}=0\.5; the axisymmetric source term implemented insource\_terms\_axisymmetricwith an axis regularizationr≤\\leq1\.0e\-10returning the zero source vector to avoid the1/r1/rsingularity; and a five\-symbol boundary dictionary \(:inflow\_upstream,:farfield\_outer,:outflow\_downstream,:capsule\_surface,:symmetry\_axis\) matched to the mesh tagging\. Pseudo\-time integration runs totfinal=200t\_\{\\mathrm\{final\}\}=200with a residual\-drop early\-stop criterion of three orders of magnitude in the entropy\-distance proxy \(achieved att≈4\.82t\\approx 4\.82in the realized run\)\.

#### B\.1\.5Stage 11 — postprocessing and validation diagnostics

The postprocessing stage selects four representative snapshots att≈\{0\.50,1\.50,3\.00,5\.00\}t\\approx\\\{0\.50,\\ 1\.50,\\ 3\.00,\\ 5\.00\\\}\(loaded by nearest\-time match, no temporal interpolation\) and three diagnostic fields per snapshot\. The agent’s nomination of the Mach diagnostic is explicit,

> “Mach numberM=\|u\|/aM=\|u\|/awitha=γ​p/ρa=\\sqrt\{\\gamma p/\\rho\}\. On every panel, overplot theM=1M=1contour in black to delineate the sonic surface — watching this contour move outward from the capsule between snapshots is the clearest visual of bow\-shock stand\-off growth\.” Postprocessing\_Details\.md, lines 110–114

\(Fig\.[4](https://arxiv.org/html/2605.11117#S2.F4)D renders theM=1M=1contour in red rather than black for visibility against the dark\-blue subsonic pocket; the contour itself is the agent’s specification\.\) The realized stand\-off evolution along the stagnation streamline \(computed from the per\-snapshot maximum of\|∇ρ\|\|\\nabla\\rho\|inr≈0r\\approx 0,z∈\[z∞,0\]z\\in\[z\_\{\\infty\},0\]\) isΔ​\(t\)≈\{0\.49,1\.37,2\.00,2\.00\}​Rb\\Delta\(t\)\\approx\\\{0\.49,\\ 1\.37,\\ 2\.00,\\ 2\.00\\\}\\,R\_\{b\}, stabilizing at the inflow\-bounded value byt=3t=3\.

#### B\.1\.6Stages 12–13 — diagnostic verdict

The diagnostic stage screens the realized solution against nine canonical failure modes for numerical CFD runs \(visualization failure, solver crash, non\-finite fields, entropy instability, feature smearing, oscillation, asymmetry, positivity warning, anomaly\)\. The recorded verdict for this run is that all nine returned “Not observed”, with integrity score25/2525/25and details score15/1515/15\. The advisor stage then evaluated the full method, mesh, and implementation against the run outcome and prescribed no edits,

> “Method already at ceiling — no advice needed\.” advisor\_report\.txt, line 47

giving a final reward breakdown of\{Accuracy,Integrity,Details,Efficiency,Optimality\}=\{25,25,15,20,15\}\\\{\\mathrm\{Accuracy\},\\mathrm\{Integrity\},\\mathrm\{Details\},\\mathrm\{Efficiency\},\\mathrm\{Optimality\}\\\}=\\\{25,25,15,20,15\\\}for a total of100/100100/100, status APPROVED\.

#### B\.1\.7Single\-pass record

The run was completed in a single iteration\.Problem\_history\.mdrecords “iterations recorded: 1” with status APPROVED\. The method\-refinement log is a single decision path through the nine\-level tree, not a sequence of retries;proposed\_method\.jsonis the final method, with no prior accepted version\. The advisor returned no mesh edits, no implementation edits, and no method edits\. No solver crash, no checkpoint resumption, and no manual intervention were recorded between problem submission and the final approved figure\. The eight numerical decisions of Table[2](https://arxiv.org/html/2605.11117#A2.T2), the six\-zone mesh, the elixir\-level implementation choices of §[B\.1\.4](https://arxiv.org/html/2605.11117#A2.SS1.SSS4), and the validation\-diagnostic nomination of §[B\.1\.5](https://arxiv.org/html/2605.11117#A2.SS1.SSS5)are therefore all attributable to the system’s compiled action tree on this case, without iterative correction\.

### B\.2Autonomous reformulation and solution of a perivascular flow inverse problem\.

This appendix records the Formalization team trace behind the result of §[2\.7](https://arxiv.org/html/2605.11117#S2.SS7)\. The case is the inverse reconstruction of velocity and pressure fields in the perivascular space of a mouse from sparse two\-photon microscopy tracks under the moving\-boundary formulation ofToscanoet al\.\[[2024](https://arxiv.org/html/2605.11117#bib.bib194)\]\. The complete artefacts \(exact\-solution check, simplification proposals, ranking, derivation, well\-posedness audit, user discussion log\) are archived in the run directoryPerivascular\_Stokes\_3D\_20260428, subfoldermathematical\_analysis/; verbatim quotations below are drawn from those documents and cited inline by file\. The final verdict wasconditionally\_posed, well\-posed up to the residual 2\-potential gaugeχ​\(x,z,t\)\\chi\(x,z,t\), after four flagged deficits were closed \(two by direct agent reasoning, two by user interaction\)\.

#### B\.2\.1Stage 1 — exact\-solution check

The proposer first searched for closed\-form analytical solutions on the full system\. The result was negative,

> “No exact analytical method identified\.” Exact\.md

the irregular moving\-wall geometry, unknown initial condition, scattered boundary data, and inverse reconstruction goal jointly precluding a closed\-form solution\.

#### B\.2\.2Stage 2 — simplification proposals

Simplify\.mdreturned three candidates\. The first is a vector\-potential reformulation of the velocity,

> “Use a vector\-potential representation of the velocity as a simplification\-only method\. The unknown velocity field is written as the curl of a vector potential, which enforces incompressibility identically\.” Simplify\.md, Case 1

The second is a pressure\-Poisson and Leray\-projection reduction, which separates pressure from the solenoidal velocity evolution by projecting the momentum equation onto divergence\-free fields\. The third is a boundary\-fitted coordinate transform mappingΩ​\(t\)\\Omega\(t\)onto a fixed reference domainΩ^\\widehat\{\\Omega\}via𝐱=𝐗​\(𝝃,t\)\\mathbf\{x\}=\\mathbf\{X\}\(\\boldsymbol\{\\xi\},t\), absorbing the geometric complexity into the deformation gradientFF, the JacobianJJ, and the grid velocity𝐚\\mathbf\{a\}\.

#### B\.2\.3Stage 3 — ranking

The ranker scored each candidate against the six constraints of §[4\.2\.1](https://arxiv.org/html/2605.11117#S4.SS2.SSS1), augmented by a hard\-constraint coverage axis\. C1 was first promoted from the soft\-Coulomb\-gauge 3\-component form to a 2\-potential specialization,

> “Simplify\.mdCase 1 proposes a 3\-component vector potential𝐀=\(A1,A2,A3\)\\mathbf\{A\}=\(A\_\{1\},A\_\{2\},A\_\{3\}\)with a soft Coulomb gauge∇⋅𝐀=0\\nabla\\cdot\\mathbf\{A\}=0enforced as an additional residual\. The skill rubric explicitly discourages that form \(it gives back the residual it eliminated\)\. The canonical PINN form is the 2\-potential specializationA2≡0A\_\{2\}\\equiv 0: the network outputs two scalar potentials\(P,R\)\(P,R\)instead of\(u,v,w\)\(u,v,w\), and the velocity is the curlu=Ryu=R\_\{y\},v=Pz−Rxv=P\_\{z\}\-R\_\{x\},w=−Pyw=\-P\_\{y\}\. Continuity … then holds as a vector\-calculus identity — no residual is needed for it\.” Ranking\.md, Candidate Summaries C1

C2 was disqualified for primitive\-PINN purposes,

> “Mathematically clean, but in a primitive\-variable PINN this is operationally inert: the scalar residuals it produces are identical to those of the original primitive Stokes formulation, and the only ‘new’ content is an auxiliaryΔ​p=0\\Delta p=0residual that adds a residual rather than removing one\. The Leray projection itself is not a clean autodiff operation — applying it pointwise requires solving an auxiliary elliptic problem at every collocation point\.” Ranking\.md, Candidate Summaries C2

and C3 was disqualified for missing geometry inputs,

> “The problem ships dense geometry\-respecting visualization points and scattered moving\-wall samples — neither is a centerline, structured mesh, or analytic parameterization\. Constructing𝐗\\mathbf\{X\}is a separate geometry pipeline; locating each sparse observation\(xi,yi,zi\)\(x\_\{i\},y\_\{i\},z\_\{i\}\)in reference coordinates requires numerically inverting𝐗\\mathbf\{X\}, an auxiliary nonlinear problem at every observation point\.” Ranking\.md, Why C3 lost

The final scores under the rubricS=0\.25​DimRed\+0\.20​NonlinRed\+0\.15​Reg\+0\.15​HardCnstr−0\.10​BCCost−0\.10​ImplCost\+0\.05​CompS=0\.25\\,\\text\{DimRed\}\+0\.20\\,\\text\{NonlinRed\}\+0\.15\\,\\text\{Reg\}\+0\.15\\,\\text\{HardCnstr\}\-0\.10\\,\\text\{BCCost\}\-0\.10\\,\\text\{ImplCost\}\+0\.05\\,\\text\{Comp\}are summarized in Table[3](https://arxiv.org/html/2605.11117#A2.T3)\.

Table 3:Ranking of the three simplification candidates\.Only C1 lies above the0\.100\.10meaningful\-benefit threshold; the load\-bearing axes are hard\-constraint coverage \(continuity becomes an identity\) and composability \(the curl reparameterization composes with output ansätze, Fourier features, and RBA / NTK weighting\)\. Both non\-winners pay full implementation cost without any compensating benefit\.The recorded justification for selecting C1 reads,

> “C1 is the only candidate that changes the operational PINN loss in a strictly favorable direction\. Network outputs drop from 4\(u,v,w,p\)\(u,v,w,p\)to 3\(P,R,p\)\(P,R,p\), the continuity residual is replaced by the vector\-calculus identity∇⋅\(∇×𝐀\)≡0\\nabla\\cdot\(\\nabla\\times\\mathbf\{A\}\)\\equiv 0, and the velocity Dirichlet conditions on the moving wall and the sparse\(u,v\)\(u,v\)observations remain Dirichlet — they just act on linear combinations of first derivatives of\(P,R\)\(P,R\)instead of on the primitive components\.” Ranking\.md, Why the winner won

#### B\.2\.4Stage 4 — derivation

Derivation\.mdcarried out the 2\-potential substitution\. SettingP:=A1P:=A\_\{1\},R:=A3R:=A\_\{3\}under the gaugeA2≡0A\_\{2\}\\equiv 0gives

u=∂yR,v=∂zP−∂xR,w=−∂yP,u=\\partial\_\{y\}R,\\qquad v=\\partial\_\{z\}P\-\\partial\_\{x\}R,\\qquad w=\-\\partial\_\{y\}P,and the continuity residual collapses identically:

∂xu\+∂yv\+∂zw=∂x​yR\+∂y​zP−∂x​yR−∂y​zP≡0\.\\partial\_\{x\}u\+\\partial\_\{y\}v\+\\partial\_\{z\}w=\\partial\_\{xy\}R\+\\partial\_\{yz\}P\-\\partial\_\{xy\}R\-\\partial\_\{yz\}P\\equiv 0\.The headline of the resulting PINN reformulation is recorded as,

> “Replace the four primitive unknowns\(u,v,w,p\)\(u,v,w,p\)by three new outputs\(P,R,p\)\(P,R,p\), with\(u,v,w\)=\(Ry,Pz−Rx,−Py\)\(u,v,w\)=\(R\_\{y\},\\ P\_\{z\}\-R\_\{x\},\\ \-P\_\{y\}\)\. Continuity∇⋅𝐮=0\\nabla\\cdot\\mathbf\{u\}=0then holds as a vector\-calculus identity, so its residual is removed from the PINN loss\. The momentum equations remain three scalar PDEs in the new outputs, with one extra derivative order on\(P,R\)\(P,R\)in the viscous term\.” Derivation\.md, Overview

The momentum residuals become

∂t\(∂yR\)\+∂xp−Δ​\(∂yR\)=0,\\partial\_\{t\}\(\\partial\_\{y\}R\)\+\\partial\_\{x\}p\-\\Delta\(\\partial\_\{y\}R\)=0,∂t\(∂zP−∂xR\)\+∂yp−Δ​\(∂zP−∂xR\)=0,\\partial\_\{t\}\(\\partial\_\{z\}P\-\\partial\_\{x\}R\)\+\\partial\_\{y\}p\-\\Delta\(\\partial\_\{z\}P\-\\partial\_\{x\}R\)=0,∂t\(−∂yP\)\+∂zp−Δ​\(−∂yP\)=0,\\partial\_\{t\}\(\-\\partial\_\{y\}P\)\+\\partial\_\{z\}p\-\\Delta\(\-\\partial\_\{y\}P\)=0,the network output count drops from four to three, the residual count drops from four to three, and the moving\-wall and sparse\-observation Dirichlet conditions are re\-expressed as Dirichlet residuals on linear combinations of first derivatives of\(P,R\)\(P,R\)\.

#### B\.2\.5Stage 5 — well\-posedness audit

The well\-posedness agent ran an eight\-axis audit covering PDE classification, BC and IC counting, null\-space and gauge detection, energy and coercivity, identifiability, continuous dependence on data, observation support, and PIML\-specific concerns\. The initial verdict wasconditionally\_posed, with four deficits flagged: two were closed by direct agent reasoning \(D1, D3\); two through interaction with the user \(D2, D4\)\.

##### D1 — pressure additive gauge\.

The reformulation inherits the gaugep↦p\+c​\(t\)p\\mapsto p\+c\(t\)from primitive Stokes; every residual is invariant under this shift, soc​\(t\)c\(t\)lives in the null space of the loss\. The agent inferred channel orientation from the data \(wall samples havevb≡0v\_\{b\}\\equiv 0, indicatingyyis the channel\-axial direction; the visualization grid is uniformly distributed alongyy; wall samples do not cover the bounding\-box faces aty=0\.011y=0\.011andy=1\.556y=1\.556\) and prescribed an outlet anchor,

> “Add a Dirichlet pressure boundary conditionp​\(t,x,ymax,z\)=0p\(t,x,y\_\{\\max\},z\)=0on the outlet end\-cap of the channel, defined as the cross\-section atymax=1\.556y\_\{\\max\}=1\.556\. … Imposingp​\(t,x,1\.556,z\)=0p\(t,x,1\.556,z\)=0at everyttremoves the additive gaugec​\(t\)c\(t\)entirely \(the boundary trace forcesc​\(t\)≡0c\(t\)\\equiv 0\)\.” well\_posedness\_report\.md, Deficit D1 resolution

##### D2 — single\-slab observability\.

Interior\(uobs,vobs\)\(u\_\{\\rm obs\},v\_\{\\rm obs\}\)data lie on the slabz∈\[0\.40,0\.42\]z\\in\[0\.40,0\.42\], covering≈2\.3%\\approx 2\.3\\%of thezz\-extent\. The user opened the exchange,

> “wait we only have data in a single slice at z around the middle of the channel is that enough?” Discussion\.md, user, 2026\-04\-28 14:55 UTC

The agent acknowledged the identifiability gap,

> “The forward Stokes problem with full Dirichlet wall data \+ outlet pressure \+ IC is uniquely solvable\. With the IC unknown and only one observation slab, there is a real identifiability gap: any IC mode that has a node at the slab planez∈\[0\.40,0\.42\]z\\in\[0\.40,0\.42\]produces zero observable\(u,v\)\(u,v\)on the slab and is invisible to the data residual\. Such modes … decay parabolically at rate∝e−λ2​t\\propto e^\{\-\\lambda^\{2\}t\}but persist over the early\-time window\.” Discussion\.md, agent reply

and offered three closures: \(A\) keep the current plan with stratified validation only, \(B\) add a smoothness prior on∂z2\\partial\_\{z\}^\{2\}of the velocity that suppresses the high\-frequency unobservable IC modes, or \(C\) trim the early\-time window where the IC dominates\. The user chose B,

> “lets do B …” Discussion\.md, user follow\-up

and the agent added the regularizer

ℒreg=λreg​𝔼​\[\|∂z2u\|2\+\|∂z2v\|2\+\|∂z2w\|2\],\\mathcal\{L\}\_\{\\rm reg\}=\\lambda\_\{\\rm reg\}\\,\\mathbb\{E\}\\\!\\left\[\\,\|\\partial\_\{z\}^\{2\}u\|^\{2\}\+\|\\partial\_\{z\}^\{2\}v\|^\{2\}\+\|\\partial\_\{z\}^\{2\}w\|^\{2\}\\right\],acting on the velocity \(gauge\-invariant for both null modes; constant and linearzz\-profiles unbiased\) together with a stratified validation protocol that reports near\-slab and far\-from\-slab errors separately\.

##### D3 — unknown initial condition\.

The IC is recognized as a formulation choice \(inverse / data\-assimilation\), not a deficit; wall and observation residuals att=0t=0act as the IC closure and no additional structure is required\.

##### D4 — domain topology\.

The 2\-potential ansatz withA2≡0A\_\{2\}\\equiv 0requires the channel to be simply connected; otherwise harmonic correction terms enter the representation\. The agent asked the user to confirm the topology, and the resolution was recorded as,

> “User confirmed the geometry is a single open channel — no bifurcation, no through\-holes,b1=0b\_\{1\}=0\. The 2\-potential ansatz𝐮=∇×\(P,0,R\)\\mathbf\{u\}=\\nabla\\times\(P,0,R\)is therefore structurally complete on this domain; no harmonic\-correction term is required\.” well\_posedness\_report\.md, Deficit D4 resolution

#### B\.2\.6Final verdict

After the four closures, the audit upgrades to,

> “conditionally\_posed— well\-posed up to the residual 2\-potential gaugeχ​\(x,z,t\)\\chi\(x,z,t\)\. … The only remaining null mode in the formulation is the 2\-potential residual gaugeχ​\(x,z,t\)\\chi\(x,z,t\), which is observably zero \(does not enter the velocity field\) and is intentionally left free perDerivation\.md\. The formulation is therefore well\-posed for every observable quantity \(velocity field, pressure field\) and conditionally posed in the strict sense — well\-posed up to that benign residual gauge\.” well\_posedness\_report\.md, post\-closure verdict

The reformulated problem then enters the encode\-select\-solve spine of §[4\.1\.8](https://arxiv.org/html/2605.11117#S4.SS1.SSS8)\.

### B\.3Agent\-designed spectral PINN for viscous Burgers\.

This appendix records the Formalization team trace behind the result of §[2\.8](https://arxiv.org/html/2605.11117#S2.SS8)\. The case is the periodic viscous Burgers equationut\+u​ux−ν​ux​x=0u\_\{t\}\+uu\_\{x\}\-\\nu u\_\{xx\}=0on\(t,x\)∈\(0,1\]×\[−1,1\]\(t,x\)\\in\(0,1\]\\times\[\-1,1\]withν=1/100\\nu=1/100and ICu​\(0,x\)=−sin⁡\(π​x\)u\(0,x\)=\-\\sin\(\\pi x\)\. The complete artefacts \(exact\-solution check, simplification proposals, ranking with user override, derivation with three review\-driven repair cycles, well\-posedness audit, parity\-fold report\) are available on the associated repository\. Verbatim quotations below are drawn from those documents and cited inline by file\. The interaction trace shown here is the one referenced in the second sentence of §[2\.8](https://arxiv.org/html/2605.11117#S2.SS8): the rubric initially demoted the spectral\-Galerkin candidate as a high\-implementation\-risk research bet, and the user override flipped it back; the agent then reasoned out, on its own, why the rubric had been mis\-scoring the problem\.

#### B\.3\.1Stage 1 — exact\-solution check

The proposer searched for closed\-form solutions on the full equation\. The result was negative,

> “Not allowed by user \(exact\_allowedunset; default deny\)\. Method present in the wiki \(Cole–Hopf linearization\) but excluded per the skill default\.” Exact\.md, summarized inRanking\.md

so the canonical Cole–Hopf route was logged for traceability and removed from the candidate pool\.

#### B\.3\.2Stage 2 — simplification proposals

The proposer agent generated three simplification candidates and recorded them inSimplify\.md: an inviscid high\-Reynolds\-number outer limit, a periodic Fourier\-series mode reduction, and a conservative flux\-divergence reformulation\. The Fourier\-series candidate is the one that becomes the spectral PINN of §[2\.8](https://arxiv.org/html/2605.11117#S2.SS8), and the agent had already identified its structural payoff at the proposal stage,

> “The spatial domain is periodic with period22, so the natural basis isei​n​π​xe^\{in\\pi x\}forn∈ℤn\\in\\mathbb\{Z\}\. … The Fourier transform removes the continuous spatial coordinate and produces a countable system of time\-dependent ODEs for modal coefficients\.” Simplify\.md, Case 2 \(agent proposal\)

The same file flagged the inviscid limit as regime\-incompatible because the characteristic map loses invertibility at the IC\-determined timets=1/π≈0\.318t\_\{s\}=1/\\pi\\approx 0\.318and “cannot represent the viscous internal layer that defines the problem\.” At the ranking stage the ranker added a fourth candidate of its own, multi\-harmonic Fourier\-feature input encoding, as a near\-zero\-cost alternative to the spectral\-Galerkin proposal it was about to score; this is the candidate that ended up displacing the proposer’s Fourier\-series suggestion in the rubric verdict\.

#### B\.3\.3Stage 3 — ranking and user override

The ranker scored each candidate against the same six\-axis rubric used in §[B\.2\.3](https://arxiv.org/html/2605.11117#A2.SS2.SSS3)\. The verdict placed C4 \(multi\-harmonic Fourier\-feature input encoding\) at rank 1 and C2 \(spectral\-Galerkin mode reduction\) at rank 2, on the grounds that C4 buys hard periodicity at near\-zero implementation cost while C2 inflates the network output to2​N2Nmodes, adds a dealiased pseudospectral evaluation per residual, and introduces ODE stiffness\. The user overrode this verdict and selected C2,

> “Selected method: C2, Pseudospectral Fourier–Galerkin Mode Reduction \(Spectral PINN\)\. The rubric below ranks C4 \(multi\-harmonic Fourier\-feature input encoding\) first on a ‘minimal architectural delta from a standard PINN’ basis\. The user has explicitly overridden this and selected C2 as the winner\.” Ranking\.md, User Override

The agent then reasoned out, on its own, why the rubric had under\-scored C2 against the structure of this specific problem,

> “What the rubric does not score\. Diagonal viscous damping: C2’s−ν​\(n​π\)2\-\\nu\(n\\pi\)^\{2\}term is closed\-form, not learned; the hardest part of training a PINN on Burgers, getting the MLP to faithfully representν​ux​x\\nu u\_\{xx\}near the layer, is removed in C2, while C4 still has to learn it\. Trivial IC:u​\(0,x\)=−sin⁡\(π​x\)u\(0,x\)=\-\\sin\(\\pi x\)is exactly theb1​sin⁡\(π​x\)b\_\{1\}\\sin\(\\pi x\)mode in C2, and under hard parameterizationb1​\(t\)=−1\+t​b~1​\(t\)b\_\{1\}\(t\)=\-1\+t\\tilde\{b\}\_\{1\}\(t\)the IC is structural \(no residual\), while C4 still carries an IC residual\. Spatial autodiff removed: C2’s PINN has no∂x,∂x​x\\partial\_\{x\},\\partial\_\{xx\}to differentiate through, those are spectral multiplications, while C4 still doesu^x,u^x​x\\hat\{u\}\_\{x\},\\hat\{u\}\_\{xx\}via autodiff at every collocation point\. Conservation laws structural: mean conservation in C2 is enforced by omittinga0a\_\{0\}from the network output, while C4 only gets it approximately through training\.” Ranking\.md, “Why C4 was rubric\-ranked first \(and why the rubric isn’t the whole story\)”

The same passage flagged a self\-correction the agent had performed against an earlier draft of its own rubric scoring,

> “Correction to an earlier composability claim\. A previous draft of this ranking asserted that ‘standard pointwise PIML weighting recipes \(RBA / NTK / soft\-attention\) do not transfer mechanically to per\-mode reweighting\.’ This was wrong\. RBA, NTK reweighting, and soft\-attention are all per\-residual\-evaluation schemes, they don’t care whether the second tensor axis indexes spatial collocation or mode index\. … The composability score of0\.300\.30in the rubric table above is therefore overly pessimistic; a fairer score would be≥0\.70\\geq 0\.70\(which would push C2 above C4 on the rubric, but the rubric weights are kept as\-is for audit\-trail transparency\)\.” Ranking\.md, “Why each non\-winner lost” \(rubric verdict, kept for the record\)

so the user override and the agent’s own re\-scoring of composability point in the same direction; both were folded into the live formulation that proceeded to derivation\.

#### B\.3\.4Stage 4 — derivation and three review\-driven repairs

Derivation\.mdcarried out the spectral\-Galerkin reduction\. Expanding

u​\(t,x\)=∑n=1N\[an​\(t\)​cos⁡\(n​π​x\)\+bn​\(t\)​sin⁡\(n​π​x\)\]u\(t,x\)=\\sum\_\{n=1\}^\{N\}\[a\_\{n\}\(t\)\\cos\(n\\pi x\)\+b\_\{n\}\(t\)\\sin\(n\\pi x\)\], projecting the PDE onto the basis, and parameterizing the modal coefficients by hard\-IC encodingan​\(t\)=t​a~n​\(t\)a\_\{n\}\(t\)=t\\tilde\{a\}\_\{n\}\(t\),bn​\(t\)=bnI​C\+t​b~n​\(t\)b\_\{n\}\(t\)=b\_\{n\}^\{IC\}\+t\\tilde\{b\}\_\{n\}\(t\)withb1I​C=−1b\_\{1\}^\{IC\}=\-1produces the coupled2​N2N\-residual ODE system ont∈\(0,1\]t\\in\(0,1\]\. The pseudospectral nonlinearityu​uxuu\_\{x\}is evaluated on a dealias grid ofMMpoints and projected back onto theNNretained modes; the viscous term is diagonal in modal space\. The first version of the derivation, however, did not survive review,

> “Overall verdict: partially correct but needs major repair\. … The viscous term has the wrong sign in the final real\-mode residuals\. …Derivation\.mdinstead writes−ε​\(n​π\)2​an\-\\varepsilon\(n\\pi\)^\{2\}a\_\{n\}and−ε​\(n​π\)2​bn\-\\varepsilon\(n\\pi\)^\{2\}b\_\{n\}, which would produce anti\-diffusive modal growth rather than viscous damping when solved asa˙n=−𝒩na−ε​\(n​π\)2​an\\dot\{a\}\_\{n\}=\-\\mathcal\{N\}\_\{n\}^\{a\}\-\\varepsilon\(n\\pi\)^\{2\}a\_\{n\}\.” review\_1\_derivation\.md

The reviewer also caught an algebra typo in the time derivative of the hard\-IC sine coefficient \(b˙n=b~n\+t​b~˙n\\dot\{b\}\_\{n\}=\\tilde\{b\}\_\{n\}\+t\\dot\{\\tilde\{b\}\}\_\{n\}, notb~˙n\+t​b~˙n\\dot\{\\tilde\{b\}\}\_\{n\}\+t\\dot\{\\tilde\{b\}\}\_\{n\}\) and pushed back on the dealias prescription\. The next pass repaired the sign and the derivative, and the second review then sharpened the dealias rule fromM≥3​NM\\geq 3Nto a strict inequality,

> “With retained complex modes\|k\|≤N\|k\|\\leq N, the productu​uxuu\_\{x\}contains modes up to\|k\|≤2​N\|k\|\\leq 2N\. A discrete grid of lengthM=3​NM=3Ncan alias the product mode2​N2Ninto the retained mode−N\-Nand the product mode−2​N\-2NintoNN\. To avoid aliasing into the retained band\|n\|≤N\|n\|\\leq N, requireM\>3​NM\>3N\.” review\_2\_derivation\.md

Three further review cycles closed a stale viscosity\-regime sweep \(ε=1/\(100​π\)→ν=1/100\\varepsilon=1/\(100\\pi\)\\to\\nu=1/100, propagated fromUser\_Question\.mdinto the recommendedNNtable\) and a stale validation\-data path that had survived in the inverse\-transform section after the data swap; the changelog of the six derivation snapshots is recorded inOld\_Derivations/CHANGELOG\.md\.

#### B\.3\.5Stage 5 — well\-posedness audit

The well\-posedness agent ran the same eight\-axis audit as in §[B\.2\.5](https://arxiv.org/html/2605.11117#A2.SS2.SSS5)\. The PDE is semi\-linear parabolic with strictly positive viscosity, BC and IC counting are exact \(periodicity is structural in the Fourier basis, the IC is structural under the hard parameterization\), the energy estimate12​dd​t​‖u‖22=−ν​‖ux‖22≤0\\tfrac\{1\}\{2\}\\tfrac\{d\}\{dt\}\\\|u\\\|\_\{2\}^\{2\}=\-\\nu\\\|u\_\{x\}\\\|\_\{2\}^\{2\}\\leq 0transfers cleanly to the spectral truncation by Parseval, and Hadamard continuity follows from standard parabolic theory\. The audit returnedwell\_posedwith two architectural observations, both closed in dialogue with the user\.

##### O1 — parity over\-parameterization\.

The ICu​\(0,x\)=−sin⁡\(π​x\)u\(0,x\)=\-\\sin\(\\pi x\)is parity\-odd, the PDE preserves odd parity under\(x,u\)→\(−x,−u\)\(x,u\)\\to\(\-x,\-u\), and the periodic parabolic IBVP has a unique solution; thereforeu​\(t,−x\)≡−u​\(t,x\)u\(t,\-x\)\\equiv\-u\(t,x\)for alltt, which forces the cosine modesan​\(t\)≡0a\_\{n\}\(t\)\\equiv 0\. The agent’s reading of the architectural consequence was,

> “This is not a well\-posedness deficit, the solution is still uniquely determined and the residuals do drivean→0a\_\{n\}\\to 0on convergence\. But: the output dimension is2​N2NwhenNNwould suffice \(a 50% architectural waste\); the cosine residuals contribute to the loss budget alongside the load\-bearing sine residuals; … residual minimization yields\|an\|∼loss/Nt\|a\_\{n\}\|\\sim\\sqrt\{\\text\{loss\}/N\_\{t\}\}, not exactly zero\.” well\_posedness\_report\.md, Observation O1

The proposed closure was a parameterization that fixesan≡0a\_\{n\}\\equiv 0structurally,

u^​\(t,x\)=∑n=1Nbn​\(t\)​sin⁡\(n​π​x\),bn​\(t\)=bnI​C\+t​b~n​\(t\),fθ:\[0,1\]→ℝN,\\hat\{u\}\(t,x\)=\\sum\_\{n=1\}^\{N\}b\_\{n\}\(t\)\\sin\(n\\pi x\),\\qquad b\_\{n\}\(t\)=b\_\{n\}^\{IC\}\+t\\,\\tilde\{b\}\_\{n\}\(t\),\\qquad f\_\{\\theta\}:\[0,1\]\\to\\mathbb\{R\}^\{N\},binding the architecture to parity\-odd ICs only\. The user confirmed the benchmark is single\-IC and approved the fold,

> “Resolution \(2026\-05\-05\)\. User confirms this is a single\-IC benchmark and approves the parity fold\. … Closure check: parity\-odd subspace is closed under the dynamics,u​ux=\(odd\)⋅\(even\)=odduu\_\{x\}=\(\\text\{odd\}\)\\cdot\(\\text\{even\}\)=\\text\{odd\}, so𝒩na=⟨u​ux,cos⁡\(n​π​x\)⟩≡0\\mathcal\{N\}\_\{n\}^\{a\}=\\langle uu\_\{x\},\\cos\(n\\pi x\)\\rangle\\equiv 0, and the cosine residualsRnaR\_\{n\}^\{a\}are identically zero in this subspace\. … Live residual count:NNscalar ODEsRnbR\_\{n\}^\{b\}\(down from2​N2N\)\.” well\_posedness\_report\.md, O1 resolution

##### O2 — single\-point smoothness gauge att=0t=0\.

Because the time\-collocation set\{ti\}⊂\(0,1\]\\\{t\_\{i\}\\\}\\subset\(0,1\]excludest=0t=0, no residual directly pins the network valuesa~n​\(0\),b~n​\(0\)\\tilde\{a\}\_\{n\}\(0\),\\tilde\{b\}\_\{n\}\(0\)\. The agent took the limit of the residuals ast→0\+t\\to 0^\{\+\}and read off the consistent values from the IC,

> “Rna​\(t\)→a~n​\(0\)\+𝒩na​\(0\)\+0=0⇒a~n​\(0\)=−𝒩na​\(0\)R\_\{n\}^\{a\}\(t\)\\to\\tilde\{a\}\_\{n\}\(0\)\+\\mathcal\{N\}\_\{n\}^\{a\}\(0\)\+0=0\\Rightarrow\\tilde\{a\}\_\{n\}\(0\)=\-\\mathcal\{N\}\_\{n\}^\{a\}\(0\), and similarlyb~n​\(0\)=−𝒩nb​\(0\)−ν​\(n​π\)2​bnI​C\\tilde\{b\}\_\{n\}\(0\)=\-\\mathcal\{N\}\_\{n\}^\{b\}\(0\)\-\\nu\(n\\pi\)^\{2\}b\_\{n\}^\{IC\}\. The IC nonlinearity isu​ux\|t=0=−sin⁡\(π​x\)⋅\(−π​cos⁡\(π​x\)\)=\(π/2\)​sin⁡\(2​π​x\)uu\_\{x\}\|\_\{t=0\}=\-\\sin\(\\pi x\)\\cdot\(\-\\pi\\cos\(\\pi x\)\)=\(\\pi/2\)\\sin\(2\\pi x\), a puren=2n=2sine mode, so the consistent values area~n​\(0\)=0​∀n\\tilde\{a\}\_\{n\}\(0\)=0\\;\\forall n,b~1​\(0\)=\+ν​π2≈0\.0987\\tilde\{b\}\_\{1\}\(0\)=\+\\nu\\pi^\{2\}\\approx 0\.0987,b~2​\(0\)=−π/2≈−1\.5708\\tilde\{b\}\_\{2\}\(0\)=\-\\pi/2\\approx\-1\.5708,b~n​\(0\)=0​\(n≥3\)\\tilde\{b\}\_\{n\}\(0\)=0\\;\(n\\geq 3\)\. This is a single\-point gauge resolved by smoothness\.” well\_posedness\_report\.md, Axis C, item 5

The closure space offered two equivalent options, addingt=0t=0to the collocation set or committing the consistent values into a smallL2L^\{2\}regularizer; both depend on the first MLP layer being smooth intt\. The user committed to a tanh activation,

> “Resolution \(2026\-05\-05\)\. User confirms the network will use tanh activations \(smooth,C∞C^\{\\infty\}\)\. Smooth activations reach the consistent smooth limit att=0\+t=0^\{\+\}by continuity ofa~n,b~n\\tilde\{a\}\_\{n\},\\tilde\{b\}\_\{n\}, so the single\-point gauge is benign without further intervention\. No additional residual att=0t=0is needed\.” well\_posedness\_report\.md, O2 resolution

#### B\.3\.6Final verdict

After both closures, the audit upgrades to,

> “well\_posed— closures applied: O1 \(parity fold\), O2 \(smooth\-activation requirement\)\. … The live formulation passes every well\-posedness axis cleanly with the closures folded in; the parameterization is now minimal \(no soft\-vs\-structural symmetry mismatch, no architectural slack\), and the loss carriesN⋅NtN\\cdot N\_\{t\}residual evaluations down from2​N⋅Nt2N\\cdot N\_\{t\}\.” well\_posedness\_report\.md, post\-closure verdict

The locked formulation, sine\-only Galerkin truncation with hard IC, hard periodic and Dirichlet structural conditions, diagonal viscous damping, and pseudospectral evaluation ofu​uxuu\_\{x\}on a dealias gridM\>3​NM\>3N, is the spectral PINN reported in §[2\.8](https://arxiv.org/html/2605.11117#S2.SS8)\(architecture in Fig\.[7](https://arxiv.org/html/2605.11117#S2.F7)\); the only active loss term is the per\-mode, per\-time vRBA\-weighted MSE of theNNsine\-Galerkin residuals\.

### B\.4Spectral PINN: implementation trace and production training\.

This appendix records the second half of the spine, the encode\-select\-solve trace from the locked formulation of §[B\.3\.6](https://arxiv.org/html/2605.11117#A2.SS3.SSS6)through code synthesis, smoke validation, production training\.

#### B\.4\.1Stage 6 — encoding through the action\-space taxonomy

The Encoding team walked the locked formulation through the action\-space taxonomy and recorded two reasoned departures from the wiki picked path\. The first concerns boundary\-condition enforcement,

> “BC: Periodic Embedding \> Periodic via Output Transformation\. The implementation MUST NOT add any output\-transformation code, nou=NN\+cos⁡\(…\)u=\\mathrm\{NN\}\+\\cos\(\\ldots\)wrapping, no distance function, no periodic embedding layer\. The structural enforcement comes from the Fourier\-sine reconstructionu^​\(t,x\)=∑n=1Nbn​\(t\)​sin⁡\(n​π​x\)\\hat\{u\}\(t,x\)=\\sum\_\{n=1\}^\{N\}b\_\{n\}\(t\)\\sin\(n\\pi x\), which is22\-periodic inxx\(and in fact zero atx=±1x=\\pm 1\) by construction\. The BC residual is empty\.” Implementation\_Details\.md, Departures

The picked\-path leaf is the wiki’s only existing terminal for “structurally enforced periodic BC,” and is reused as a label rather than a code path; no output\-side decoration enters the implementation\.

The second departure concerns the curriculum schedule\. The rubric had picked Reynolds continuation with a decaying\-coefficient schedule, and hintL1h31/L2h18flagged it as required when an internal viscous layer is present\. The agent justified skipping continuation by appealing to the spectral reformulation itself,

> “Modal stiffness scale:ν​\(N​π\)2≈0\.01⋅\(48​π\)2≈227\\nu\(N\\pi\)^\{2\}\\approx 0\.01\\cdot\(48\\pi\)^\{2\}\\approx 227atN=48N=48\. This is two orders of magnitude milder than the canonicalν=1/\(100​π\)\\nu=1/\(100\\pi\)benchmark withN=100N=100\(which givesν​\(N​π\)2≈990\\nu\(N\\pi\)^\{2\}\\approx 990\)\. The stiff feature in physical\(t,x\)\(t,x\)\-space is not stiff in\(t,n\)\(t,n\)\-space at thisNN\. … The vRBA Sampling\-only mechanism already provides a time\-axis curriculum implicitly, high\-residual time slabs aroundt≈0\.5t\\approx 0\.5are oversampled by construction\. This makes the physical effect Reynolds continuation would buy \(gradual exposure to the layer\) redundant\.” Implementation\_Details\.md, Curriculum schedule

so training proceeds at the targetν=0\.01\\nu=0\.01from iteration0of S2, with no per\-stage Hessian reinitialization and no continuation bookkeeping\.

The remainder of the encoding is a direct fill of the picked taxonomy: a time\-only3×403\\times 40MLP with tanh activation, output dimensionN=48N=48\(≈5328\\approx 5328trainable parameters, well under the SSBroyden10510^\{5\}\-parameter limit\), hard\-IC compositionbn​\(t\)=bnI​C\+t​b~n​\(t\)b\_\{n\}\(t\)=b\_\{n\}^\{IC\}\+t\\,\\tilde\{b\}\_\{n\}\(t\)withb1I​C=−1b\_\{1\}^\{IC\}=\-1, pseudospectral nonlinearity on the dealias gridM=192\>3​NM=192\>3N, and a two\-stage training pipeline \(S1 Adam for6 2506\\,250iterations, S2 SSBroyden with backtracking line\-search for20 00020\\,000outer iterations\)\. The vRBA tensorλ∈ℝN×Nt\\lambda\\in\\mathbb\{R\}^\{N\\times N\_\{t\}\}is allocated on the\(mode,time\)\(\\text\{mode\},\\text\{time\}\)grid rather than the standard\(point,time\)\(\\text\{point\},\\text\{time\}\)grid, and the Sampling face drives the time\-collocation pdf asp​\(ti\)∝∑nλn,i2p\(t\_\{i\}\)\\propto\\sum\_\{n\}\\lambda\_\{n,i\}^\{2\}between outer SSBroyden iterations\. The mode axis is fully retained at every step\.

A constraint flagged by the encoding agent then reshapes the S1 to S2 handoff,

> “At S2 with Quasi\-Newton \(SSBroyden\), only Sampling\-only is valid; the Weighting face is forbidden because non\-stationary loss multipliers corrupt the SSBroyden inner\-solveH0−1H\_\{0\}^\{\-1\}inverse\-Hessian estimate\. … In S2,λ\\lambdacontinues to update per outer iteration AND drives thesample\_points\_pdfover the\(n,i\)\(n,i\)grid for the next minibatch, but the loss expression in the SSBroyden inner solve uses unit weights\.” Implementation\_Details\.md, S2 vRBA

soλ\\lambdaruns as a Weighting\+Sampling tensor under Adam in S1 and as a Sampling\-only tensor under SSBroyden in S2, with the sameλ\\lambdastate carried across the boundary\.

#### B\.4\.2Stage 7 — code synthesis, smoke loop, and production run

The Implementation Agent translatedImplementation\_Details\.mdinto a JAX/Flax codebase by editing the inviscid\-Burgers PIML template into a spectral PINN\. The smoke loop terminated on the first try, with both reasoned departures honored verbatim in code,

> “Smoke command:main\.py \> runner\_attempt\_1\.log 2\>&1\. … Outcome: exit code 0\. Every promised array written non\-zero size\. … Health signals: SSBroyden never crashed;H0cholesky NaN guard never triggered; no instability fall\-back tobest\_weights; per\-mode max loss stayed within1\.5×1\.5\\timesof the mean throughout S2; lambda max grew monotonically as expected\.” implementation\_report\.txt, Attempt history

After the smoke caps were reverted to production values \(S1=6 250=6\\,250, S2=20 000=20\\,000\), the production run was submitted to Brown’s Oscar cluster through the canonical PIML SLURM submitter \(cascadingH100→L40S→3090→A6000\\mathrm\{H\}100\\to\\mathrm\{L\}40\\mathrm\{S\}\\to 3090\\to\\mathrm\{A\}6000\)\. The job landed on anL40S\\mathrm\{L\}40\\mathrm\{S\}and completed in70\.7170\.71s wall\-clock \(S1=9\.03=9\.03s, S2=61\.68=61\.68s\), with timing logged tohistory/timing\.json\. The training\-history archive carried a single S1→\\toS2 handoff at iter≈6 250\\approx 6\\,250, no anomalous jumps, and a per\-mode worst\-loss curve that mirrored the total loss without lagging modes\.

#### B\.4\.3Stage 8 — postprocessing and diagnostic verdict

The postprocessing layer reconstructedu^​\(t,x\)\\hat\{u\}\(t,x\)on the reference grid by analytic sum,u^dense=bdense​Sdense⊤\\hat\{u\}\_\{\\mathrm\{dense\}\}=b\_\{\\mathrm\{dense\}\}\\,S\_\{\\mathrm\{dense\}\}^\{\\top\}, and computed the strong\-form residualut\+u​ux−ν​ux​xu\_\{t\}\+uu\_\{x\}\-\\nu u\_\{xx\}on the same grid from\(u^,u^t,u^x,u^x​x\)\(\\hat\{u\},\\hat\{u\}\_\{t\},\\hat\{u\}\_\{x\},\\hat\{u\}\_\{xx\}\)via the modal expansion \(no spatial autodiff\)\. The final\-iteration metrics on the dense\(200,256\)\(200,256\)grid are

RL2u=1\.107×10−3,Lu∞=4\.91×10−3,Lu1=3\.39×10−4,\\mathrm\{RL2\}\_\{u\}=1\.107\\times 10^\{\-3\},\\qquad L^\{\\infty\}\_\{u\}=4\.91\\times 10^\{\-3\},\\qquad L^\{1\}\_\{u\}=3\.39\\times 10^\{\-4\},
and the per\-component PDE loss saturated at the single\-precision floorLPDE≈8\.3×10−13L\_\{\\mathrm\{PDE\}\}\\approx 8\.3\\times 10^\{\-13\}, a Galerkin\-loss reduction of4\.97×104×4\.97\\times 10^\{4\}\\timesbetween iteration0and the end of S2\. The RL2 panel descends from≈40%\\approx 40\\%at S1 onset, drops sharply at the S1→\\toS2 handoff, and is sub\-1%1\\%from iter≈10 000\\approx 10\\,000onwards\.

The diagnostic agent reads the gap between fp32\-floor Galerkin loss and10−310^\{\-3\}field error as the truncation tail of the retained spectrum, not a training pathology,

> “INCONSISTENT\_PLOTS: not observed:LPDE=8\.3​e−13L\_\{\\mathrm\{PDE\}\}=8\.3\\mathrm\{e\}\{\-13\}vsRL2=1\.1​e−3\\mathrm\{RL2\}=1\.1\\mathrm\{e\}\{\-3\}is the expected spectral\-truncation gap \(Galerkin loss measures onlyN=48N=48retained modes; field error includes the un\-resolved tail\)\. Plots and logs are mutually consistent under the truncated\-Galerkin lens\.” diagnostic\_report\.md, S6 failure\-mode ranking

and locates the truncation energy in the band the layer occupies,

> “Strong\-form PDE residual concentrates sharply in the layer bandt∈\[0\.3,0\.6\]t\\in\[0\.3,0\.6\], peaking∼1\\sim 1at the layer’s center timet≈0\.5t\\approx 0\.5and decaying to∼10−1\\sim 10^\{\-1\}either side\. This is the canonical signature of spectral truncation at the chosenNN: … the un\-resolved tail \(modesn\>48n\>48\) carries energy precisely where the layer’s high\-wavenumber content lives\. Modal energy\|b48\|2≈10−7\|b\_\{48\}\|^\{2\}\\approx 10^\{\-7\}, indicating modesn\>48n\>48carry non\-negligible energy\.” diagnostic\_report\.md, S4–S5

The eight\-axis failure\-mode ranking returnedSTAGNATIONatLPDEL\_\{\\mathrm\{PDE\}\}floor \(integrity25→1525\\to 15, Galerkin residual cannot be reduced further at thisNN\),SHOCK\_HIGH\_GRADIENTmild \(−4\-4, the viscous layer att≈0\.5t\\approx 0\.5is the dominant high\-gradient feature\), andSPECTRAL\_BIASmild \(−3\-3, modal energy at the truncation cutoff\)\.EXPLOSION,OSCILLATION,BOUNDARY\_MISMATCH, andVISUALIZATION\_FAILUREwere all not observed; in particular, the boundary check is exact rather than approximate becauseu^​\(t,±1\)≡0\\hat\{u\}\(t,\\pm 1\)\\equiv 0structurally by sine basis\.

The advisor closed the iteration with a single implementation prescription and no action\-tree edits, root\-causing both flagged failure modes to the same scalar knob,

> “This iteration trained the user\-locked spectral PINN cleanly to fp32 floor on the retainedN=48N=48modes \(LPDE=8\.3​e−13L\_\{\\mathrm\{PDE\}\}=8\.3\\mathrm\{e\}\{\-13\}\), but the field errorRL2u=1\.107​e−3\\mathrm\{RL2\}\_\{u\}=1\.107\\mathrm\{e\}\{\-3\}is2\.5×2\.5\\timesthe reference baseline because the Galerkin loss does not see the un\-resolved modal tail \(n\>48n\>48\)\. The remediation is a single hyperparameter bump,N=48→96N=48\\to 96,M=192→320M=192\\to 320, already named in theUser\_Request‘RecommendedNN’ progression table as the production tier and explicitly anticipated by the Implementation spec\. No action\-tree edits are warranted: the topology selections all match the user’s non\-negotiable locks, and the diagnostic does not implicate model capacity, optimizer choice, or weighting strategy\.” advisor\_report\.txt, Refinement instructions

The flatness signal is reclassified toFLAT\-on\-floor, “not a training stagnation that more iterations could break, but a truncation symptom that theN→96N\\to 96prescription resolves at the source\.” The Adam→\\toSSBroyden chain, the parity\-folded sine basis, the hard IC and structural BC, and the vRBA Sampling\-only mechanism under SSBroyden all stay unchanged; the second iteration enters the spine with the new\(N,M\)=\(96,320\)\(N,M\)=\(96,320\)pair, the precomputed sine/cosine matrices andbI​Cb^\{IC\}resized accordingly, and the SSBroyden parameter budget \(≈7\.4\\approx 7\.4k, still well under the10510^\{5\}limit\) untouched\.

#### B\.4\.4Stage 9 — second iteration: representation bump and budget verdict

The second closed\-loop iteration enters with the first iteration’s prescription applied\. TheImplementation\_Details\.mdfor this run records the bump verbatim \(“Iter\-11mode count\.N=96N=96, output dimension=96=96\. \(Advisor\-prescribed bump from iter\-0N=48N=48; permitted progression per the user formulation:32/48/96−12832/48/96\\\!\-\\\!128\.\)”\), and the corresponding pseudospectral grid choice \(M=320M=320, the smallest FFT\-friendly even integer above3​N=2883N=288that satisfies the strictM\>3​NM\>3Ndealiasing constraint\)\. No other action in the encoded chain changes\. The MLP topology is still3×403\\times 40tanh; the output dimension grows from4848to9696, lifting the parameter count from≈5\.3\\approx 5\.3k to≈7\.4\\approx 7\.4k, comfortably under the SSBroyden ceiling of10510^\{5\}MLP parameters\. The hard\-IC vectorbI​C∈ℝ96b^\{IC\}\\in\\mathbb\{R\}^\{96\}, the precomputed matricesSM,CM′∈ℝ320×96S\_\{M\},C^\{\\prime\}\_\{M\}\\in\\mathbb\{R\}^\{320\\times 96\}andSM,back∈ℝ96×320S\_\{M,\\mathrm\{back\}\}\\in\\mathbb\{R\}^\{96\\times 320\}, the diagonal viscous coefficientν​\(n​π\)2\\nu\(n\\pi\)^\{2\}forn=1,…,96n=1,\\dots,96, and the vRBA tensorλ∈ℝ96×Nt\\lambda\\in\\mathbb\{R\}^\{96\\times N\_\{t\}\}are all resized at startup; nothing else in the implementation moves\. Implementation re\-synthesis was a mechanical refactor of the iteration\-11artifact at the four locations the spec flagged \(NN,MM, the precomputed\-matrix shapes, theλ\\lambdatensor shape\), and the smoke loop returnedPASSEDat attempt11on the same L40S queue\.

The production run completed in113\.17113\.17s wall\-clock \(S1=9\.15S\_\{1\}=9\.15s,S2=104\.02S\_\{2\}=104\.02s\), against70\.7170\.71s \(9\.039\.03s,61\.6861\.68s\) atN=48N=48\. The S2 fraction rises from87%87\\%to92%92\\%of total wall as the per\-iteration cost of the residual evaluation grows with the modal tensor size; the SSBroyden inner solve absorbs the larger output dimension without backtracking instability\. The final\-iteration metrics on the dense\(200,256\)\(200,256\)grid are

RL2u=6\.598×10−6,Lu∞=4\.262×10−5,Lu1=2\.175×10−6,\\mathrm\{RL2\}\_\{u\}=6\.598\\times 10^\{\-6\},\\qquad L^\{\\infty\}\_\{u\}=4\.262\\times 10^\{\-5\},\\qquad L^\{1\}\_\{u\}=2\.175\\times 10^\{\-6\},a168×168\\timesreduction inRL2u\\mathrm\{RL2\}\_\{u\}over the first iteration’s1\.107×10−31\.107\\times 10^\{\-3\}from a single scalar knob \(NN\) and no action\-tree edit\. The diagnostic snapshot at the truncation cutoff,\|b96\|2∼10−16\|b\_\{96\}\|^\{2\}\\sim 10^\{\-16\}, confirms that theN=96N=96basis spans the layer’s wavenumber support: the\|b48\|2∼10−7\|b\_\{48\}\|^\{2\}\\sim 10^\{\-7\}tail that drove iteration\-11’s field error has fallen to the floor of the modal heatmap, and the strong\-form residual peak in the layer bandt∈\[0\.3,0\.6\]t\\in\[0\.3,0\.6\]drops by10×10\\times\. TheSPECTRAL\_BIASandSHOCK\_HIGH\_GRADIENTsignals from iteration11no longer fire\.

The diagnostic does, however, returnSTILL\_TRAINING: theRL2u\\mathrm\{RL2\}\_\{u\}trajectory is monotonically descending at the last S2 iteration with a drop fraction of0\.930\.93over the final30%30\\%window of the20 00020\\,000\-step S2 schedule \(the user\-locked iter\-11budget\)\. The second\-iteration advisor reads this as a budget\-not\-asymptote condition rather than a method failure, and prescribes the smallest preset that resolves it,

> “With the mode budget no longer a bottleneck, the residual gap is a SSBroyden\-budget issue: extending S2 to50 00050\\,000iters \(chosen as the closest preset to the diagnostic’s ‘e\.g\.20 000→40 00020\\,000\\to 40\\,000’ surfaced recommendation, and also the prior\-favoredS2T\_IT50000p=0\.160p=0\.160option\) gives the late\-S2 phase room to reach its asymptote without an aggressive5×5\\timesjump toS2T\_IT100000\. If iter\-22still firesSTILL\_TRAINING, the next bump can escalate toS2T\_IT100000\(the prior\-favored target\)\. All other selections, parity\-folded sine basis, hard IC/BC, RBA Sampling\-only under SSBroyden, Adam→\\toSSBroyden chain, stay unchanged: the method is well matched and the diagnostic localizes the deficit cleanly to the schedule\.” advisor\_report\.txt, iteration22, Refinement instructions

As at iteration11, the advisor commits no action\-tree edits and no DEP cascade: the single edit is theS2S\_\{2\}total\-iterations leaf in the optimization subtree \(S2T\_IT20000→\\toS2T\_IT50000\), and the implementation prescription propagates the new value to theS2S\_\{2\}Total iterations entry ofImplementation\_Details\.mdso that the next implementation re\-synthesis does not re\-emit the old budget\. The total budget grows from26 25026\\,250iterations \(S1S\_\{1\}24%24\\%/S2S\_\{2\}76%76\\%\) to56 25056\\,250\(S1S\_\{1\}11%11\\%/S2S\_\{2\}89%89\\%\)\.

The two\-iteration arc is the encode\-select\-solve spine working as designed\. Iteration11exposed a representation deficit \(a truncation tail the Galerkin loss could not see\), and the advisor resolved it with one scalar knob in the basis subtree\. Iteration22exposed a schedule deficit \(the asymptote was beyond the user\-locked S2 budget\), and the advisor resolved it with one scalar knob in the optimization subtree\. Neither iteration touched the topology, the optimizer family, the weighting strategy, the parity\-folded basis, or the hard IC/BC structure: every branch the Formalization\-team trace and the action\-tree encoding had locked stays locked across the loop, and the closed\-loop spine does precisely what its design promises, which is to localize the deficit at each pass to the smallest set of leaves the diagnostic supports\.

## Appendix CAgent\-built numerical\-methods action tree𝒯A\\mathcal\{T\}\_\{A\}

This appendix reproduces, as a nested list of decision chains, the numerical\-methods slice of𝒯A\\mathcal\{T\}\_\{A\}that GRAFT\-ATHENA’s Expansion and Construction teams \(§[2\.1](https://arxiv.org/html/2605.11117#S2.SS1)\) built end\-to\-end from the documentation of three production solvers: Nektar\+\+\[Cantwellet al\.,[2015](https://arxiv.org/html/2605.11117#bib.bib14)\], the Spectral Julia pseudo\-spectral stack, and Trixi\.jl\[Ranochaet al\.,[2021](https://arxiv.org/html/2605.11117#bib.bib220)\]\. We include this tree as the agent\-built artifact that backs the framework’s central claim: with it, the numerical\-side runs of §[2\.4](https://arxiv.org/html/2605.11117#S2.SS4)–§[2\.6](https://arxiv.org/html/2605.11117#S2.SS6)are reproducible end\-to\-end on a shared substrate\. The hand\-curated𝒯P\\mathcal\{T\}\_\{P\}and the PIML and DPD slices of𝒯A\\mathcal\{T\}\_\{A\}are inputs to the system rather than outputs and are not part of this release\. Sub\-attributes that hang off acc\-edge are nested under their parent, and the prior transition probability is uniform within each chain unless updated by experience\.

The tree comprises a shared Mesh chain followed by one solver\-specific subtree per family\.

### C\.1Mesh

- ∙\\bulletMS: Resolution Anisotropy - ∙\\bulletMS: Isotropic - ∙\\bulletMS: Anisotropic
- ∙\\bulletMS: Spacing - ∙\\bulletMS: Uniform Spacing - ∙\\bulletMS: Non\-uniform Spacing
- ∙\\bulletMS: Mesh Type - ∙\\bulletMS: Element\-based Mesh - ∙\\bulletMS: Element Shape - ∙\\bulletMS: Triangle - ∙\\bulletMS: Use Triangle - ∙\\bulletMS: No Triangle - ∙\\bulletMS: Quadrilateral - ∙\\bulletMS: Use Quadrilateral - ∙\\bulletMS: No Quadrilateral - ∙\\bulletMS: Tetrahedron - ∙\\bulletMS: Use Tetrahedron - ∙\\bulletMS: No Tetrahedron - ∙\\bulletMS: Hexahedron - ∙\\bulletMS: Use Hexahedron - ∙\\bulletMS: No Hexahedron - ∙\\bulletMS: Prism - ∙\\bulletMS: Use Prism - ∙\\bulletMS: No Prism - ∙\\bulletMS: Other Element Shape - ∙\\bulletMS: Use Other Element Shape - ∙\\bulletMS: No Other Element Shape - ∙\\bulletMS: Curvature - ∙\\bulletMS: Straight - ∙\\bulletMS: Curved - ∙\\bulletMS: Conformity - ∙\\bulletMS: Conformal - ∙\\bulletMS: Non\-conformal - ∙\\bulletMS: Boundary Tagging - ∙\\bulletMS: Named Edges - ∙\\bulletMS: Numeric IDs - ∙\\bulletMS: No Tags - ∙\\bulletMS: Generator - ∙\\bulletMS: Gmsh - ∙\\bulletMS: Abaqus - ∙\\bulletMS: Abaqus Straight - ∙\\bulletMS: HOHQMesh - ∙\\bulletMS: Built\-in - ∙\\bulletMS: p4est\_refine - ∙\\bulletMS: External/Unknown - ∙\\bulletMS: Collocation Grid - ∙\\bulletMS: Basis - ∙\\bulletMS: Fourier - ∙\\bulletMS: Chebyshev - ∙\\bulletMS: Legendre - ∙\\bulletMS: Hermite - ∙\\bulletMS: Other Basis - ∙\\bulletMS: Topology - ∙\\bulletMS: Periodic Box - ∙\\bulletMS: Interval - ∙\\bulletMS: Annulus - ∙\\bulletMS: Sphere - ∙\\bulletMS: Other Topology - ∙\\bulletMS: Dealiasing - ∙\\bulletMS: No Dealiasing - ∙\\bulletMS: Spectral Filter - ∙\\bulletMS: 2/3 Rule - ∙\\bulletMS: Polynomial p\-Rule - ∙\\bulletMS: Phase Shift - ∙\\bulletMS: Custom Dealiasing

### C\.2Solvers — Nektar\+\+

- ∙\\bulletNektar\_Equations - ∙\\bulletNektar\_ADR - ∙\\bulletNektar\_UnsteadyAdvection - ∙\\bulletNektar\_UnsteadyDiffusion - ∙\\bulletNektar\_UnsteadyAdvectionDiffusion - ∙\\bulletNektar\_UnsteadyReactionDiffusion - ∙\\bulletNektar\_UnsteadyInviscidBurgers - ∙\\bulletNektar\_UnsteadyViscousBurgers - ∙\\bulletNektar\_SteadyAdvectionDiffusion - ∙\\bulletNektar\_SteadyAdvectionDiffusionReaction - ∙\\bulletNektar\_Helmholtz - ∙\\bulletNektar\_Laplace - ∙\\bulletNektar\_LaplacePhi - ∙\\bulletNektar\_Poisson - ∙\\bulletNektar\_Projection\_EQ - ∙\\bulletNektar\_MMFAdvection - ∙\\bulletNektar\_EigenValuesAdvection - ∙\\bulletNektar\_INS - ∙\\bulletNektar\_VelocityCorrectionScheme - ∙\\bulletNektar\_VCSImplicit - ∙\\bulletNektar\_VCSWeakPressure - ∙\\bulletNektar\_VCSMapping - ∙\\bulletNektar\_SmoothedProfileMethod - ∙\\bulletNektar\_CoupledLinearisedNS - ∙\\bulletNektar\_Compressible - ∙\\bulletNektar\_EulerCFE - ∙\\bulletNektar\_EulerImplicitCFE - ∙\\bulletNektar\_NavierStokesCFE - ∙\\bulletNektar\_NavierStokesCFEAxisym - ∙\\bulletNektar\_NavierStokesImplicitCFE - ∙\\bulletNektar\_CardiacEP - ∙\\bulletNektar\_Monodomain - ∙\\bulletNektar\_Bidomain - ∙\\bulletNektar\_BidomainRoth - ∙\\bulletNektar\_ShallowWater - ∙\\bulletNektar\_LinearSWE - ∙\\bulletNektar\_NonlinearSWE - ∙\\bulletNektar\_NonlinearPeregrine - ∙\\bulletNektar\_MMFSWE - ∙\\bulletNektar\_PulseWave - ∙\\bulletNektar\_PulseWavePropagation - ∙\\bulletNektar\_Acoustic - ∙\\bulletNektar\_APE - ∙\\bulletNektar\_LEE - ∙\\bulletNektar\_Plasma - ∙\\bulletNektar\_DriftWaveSystem - ∙\\bulletNektar\_MMFDiffusion - ∙\\bulletNektar\_MMFMaxwell - ∙\\bulletNektar\_LinearElasticSystem - ∙\\bulletNektar\_Dummy - ∙\\bulletNektar\_FileSolution
- ∙\\bulletNektar\_Projection - ∙\\bulletNektar\_CGProjection - ∙\\bulletNektar\_DGProjection - ∙\\bulletNektar\_MixedCGDG
- ∙\\bulletNektar\_Expansions - ∙\\bulletNektar\_ExpansionType - ∙\\bulletNektar\_EXP\_Modified - ∙\\bulletNektar\_EXP\_GLLLagrange - ∙\\bulletNektar\_EXP\_GaussLagrange - ∙\\bulletNektar\_EXP\_Mixed - ∙\\bulletNektar\_BasisType - ∙\\bulletNektar\_Basis\_Modified - ∙\\bulletNektar\_Basis\_GLLLagrange - ∙\\bulletNektar\_Basis\_GaussLagrange - ∙\\bulletNektar\_PointsType - ∙\\bulletNektar\_Points\_GaussLobattoLegendre - ∙\\bulletNektar\_Points\_GaussRadauMAlpha1Beta0 - ∙\\bulletNektar\_Points\_GaussGaussLegendre - ∙\\bulletNektar\_NumModes - ∙\\bulletNektar\_Order\_p\_2 - ∙\\bulletNektar\_Order\_p\_3\-4 - ∙\\bulletNektar\_Order\_p\_5\-6 - ∙\\bulletNektar\_Order\_p\_7\-10 - ∙\\bulletNektar\_Order\_p\_gt10
- ∙\\bulletNektar\_TimeIntegration - ∙\\bulletNektar\_TI\_Euler - ∙\\bulletNektar\_TI\_RungeKutta - ∙\\bulletNektar\_TI\_AdamsBashforth - ∙\\bulletNektar\_TI\_BDFImplicit - ∙\\bulletNektar\_TI\_DIRK - ∙\\bulletNektar\_TI\_IMEX - ∙\\bulletNektar\_TI\_IMEX\_Order - ∙\\bulletNektar\_TI\_IMEX\_Order1 - ∙\\bulletNektar\_TI\_IMEX\_Order2 - ∙\\bulletNektar\_TI\_IMEX\_Order3 - ∙\\bulletNektar\_TI\_IMEX\_Order4 - ∙\\bulletNektar\_TI\_ExplicitSDC - ∙\\bulletNektar\_TI\_ImplicitSDC - ∙\\bulletNektar\_TI\_IMEXSDC - ∙\\bulletNektar\_TI\_NoScheme
- ∙\\bulletNektar\_AdvectionDiffusion - ∙\\bulletNektar\_AdvectionType - ∙\\bulletNektar\_Adv\_WeakDG - ∙\\bulletNektar\_Adv\_NonConservative - ∙\\bulletNektar\_Adv\_None - ∙\\bulletNektar\_DiffusionType - ∙\\bulletNektar\_Diff\_LDG - ∙\\bulletNektar\_Diff\_InteriorPenalty - ∙\\bulletNektar\_Diff\_LDGNS - ∙\\bulletNektar\_Diff\_None - ∙\\bulletNektar\_RiemannSolver - ∙\\bulletNektar\_Riemann\_Generic - ∙\\bulletNektar\_Riemann\_Compressible - ∙\\bulletNektar\_Riemann\_ShallowWater - ∙\\bulletNektar\_Riemann\_PulseWave - ∙\\bulletNektar\_Riemann\_Acoustic - ∙\\bulletNektar\_Riemann\_None
- ∙\\bulletNektar\_LinearSolver - ∙\\bulletNektar\_GlobalSysSoln - ∙\\bulletNektar\_GSS\_Direct - ∙\\bulletNektar\_GSS\_Iterative - ∙\\bulletNektar\_GSS\_Xxt - ∙\\bulletNektar\_GSS\_PETSc - ∙\\bulletNektar\_LinSysIterSolver - ∙\\bulletNektar\_ITER\_CG - ∙\\bulletNektar\_ITER\_GMRES - ∙\\bulletNektar\_ITER\_EvsDirect - ∙\\bulletNektar\_ITER\_NA - ∙\\bulletNektar\_Preconditioner - ∙\\bulletNektar\_PC\_Null - ∙\\bulletNektar\_PC\_Diagonal - ∙\\bulletNektar\_PC\_LowEnergyBlock - ∙\\bulletNektar\_PC\_FullLinearSpace - ∙\\bulletNektar\_PC\_LOR
- ∙\\bulletNektar\_BoundaryConditions - ∙\\bulletNektar\_BCBaseTypes - ∙\\bulletNektar\_BC\_Dirichlet - ∙\\bulletNektar\_UseBC\_Dirichlet - ∙\\bulletNektar\_NoBC\_Dirichlet - ∙\\bulletNektar\_BC\_Neumann - ∙\\bulletNektar\_UseBC\_Neumann - ∙\\bulletNektar\_NoBC\_Neumann - ∙\\bulletNektar\_BC\_Robin - ∙\\bulletNektar\_UseBC\_Robin - ∙\\bulletNektar\_NoBC\_Robin - ∙\\bulletNektar\_BC\_Periodic - ∙\\bulletNektar\_UseBC\_Periodic - ∙\\bulletNektar\_NoBC\_Periodic - ∙\\bulletNektar\_BCUserDefined - ∙\\bulletNektar\_BC\_Wall - ∙\\bulletNektar\_UseBC\_Wall - ∙\\bulletNektar\_NoBC\_Wall - ∙\\bulletNektar\_BC\_WallViscous - ∙\\bulletNektar\_UseBC\_WallViscous - ∙\\bulletNektar\_NoBC\_WallViscous - ∙\\bulletNektar\_BC\_WallAdiabatic - ∙\\bulletNektar\_UseBC\_WallAdiabatic - ∙\\bulletNektar\_NoBC\_WallAdiabatic - ∙\\bulletNektar\_BC\_RiemannInvariantBC - ∙\\bulletNektar\_UseBC\_RiemannInvariantBC - ∙\\bulletNektar\_NoBC\_RiemannInvariantBC - ∙\\bulletNektar\_BC\_IsentropicVortex - ∙\\bulletNektar\_UseBC\_IsentropicVortex - ∙\\bulletNektar\_NoBC\_IsentropicVortex - ∙\\bulletNektar\_BC\_PressureOutflow - ∙\\bulletNektar\_UseBC\_PressureOutflow - ∙\\bulletNektar\_NoBC\_PressureOutflow - ∙\\bulletNektar\_BC\_PressureOutflowNonReflective - ∙\\bulletNektar\_UseBC\_PressureOutflowNonReflective - ∙\\bulletNektar\_NoBC\_PressureOutflowNonReflective - ∙\\bulletNektar\_BC\_ExtrapOrder0 - ∙\\bulletNektar\_UseBC\_ExtrapOrder0 - ∙\\bulletNektar\_NoBC\_ExtrapOrder0 - ∙\\bulletNektar\_BC\_Symmetry - ∙\\bulletNektar\_UseBC\_Symmetry - ∙\\bulletNektar\_NoBC\_Symmetry - ∙\\bulletNektar\_BC\_RinglebFlow - ∙\\bulletNektar\_UseBC\_RinglebFlow - ∙\\bulletNektar\_NoBC\_RinglebFlow - ∙\\bulletNektar\_BC\_H - ∙\\bulletNektar\_UseBC\_H - ∙\\bulletNektar\_NoBC\_H - ∙\\bulletNektar\_BC\_HOutflow - ∙\\bulletNektar\_UseBC\_HOutflow - ∙\\bulletNektar\_NoBC\_HOutflow - ∙\\bulletNektar\_BC\_Flowrate - ∙\\bulletNektar\_UseBC\_Flowrate - ∙\\bulletNektar\_NoBC\_Flowrate - ∙\\bulletNektar\_BC\_Radiation - ∙\\bulletNektar\_UseBC\_Radiation - ∙\\bulletNektar\_NoBC\_Radiation - ∙\\bulletNektar\_BC\_MovingFrameWall - ∙\\bulletNektar\_UseBC\_MovingFrameWall - ∙\\bulletNektar\_NoBC\_MovingFrameWall - ∙\\bulletNektar\_BC\_MovingFrameFar - ∙\\bulletNektar\_UseBC\_MovingFrameFar - ∙\\bulletNektar\_NoBC\_MovingFrameFar - ∙\\bulletNektar\_BC\_MovingBody - ∙\\bulletNektar\_UseBC\_MovingBody - ∙\\bulletNektar\_NoBC\_MovingBody - ∙\\bulletNektar\_BC\_TransMovingWall - ∙\\bulletNektar\_UseBC\_TransMovingWall - ∙\\bulletNektar\_NoBC\_TransMovingWall - ∙\\bulletNektar\_BC\_Womersley - ∙\\bulletNektar\_UseBC\_Womersley - ∙\\bulletNektar\_NoBC\_Womersley - ∙\\bulletNektar\_BC\_WhiteNoise - ∙\\bulletNektar\_UseBC\_WhiteNoise - ∙\\bulletNektar\_NoBC\_WhiteNoise - ∙\\bulletNektar\_BC\_AInflow - ∙\\bulletNektar\_UseBC\_AInflow - ∙\\bulletNektar\_NoBC\_AInflow - ∙\\bulletNektar\_BC\_QInflow - ∙\\bulletNektar\_UseBC\_QInflow - ∙\\bulletNektar\_NoBC\_QInflow - ∙\\bulletNektar\_BC\_UInflow - ∙\\bulletNektar\_UseBC\_UInflow - ∙\\bulletNektar\_NoBC\_UInflow - ∙\\bulletNektar\_BC\_RTerminal - ∙\\bulletNektar\_UseBC\_RTerminal - ∙\\bulletNektar\_NoBC\_RTerminal - ∙\\bulletNektar\_BC\_RCRTerminal - ∙\\bulletNektar\_UseBC\_RCRTerminal - ∙\\bulletNektar\_NoBC\_RCRTerminal - ∙\\bulletNektar\_BC\_StagnationInflow - ∙\\bulletNektar\_UseBC\_StagnationInflow - ∙\\bulletNektar\_NoBC\_StagnationInflow - ∙\\bulletNektar\_BC\_Terminal - ∙\\bulletNektar\_UseBC\_Terminal - ∙\\bulletNektar\_NoBC\_Terminal - ∙\\bulletNektar\_BC\_TimeDependent - ∙\\bulletNektar\_UseBC\_TimeDependent - ∙\\bulletNektar\_NoBC\_TimeDependent - ∙\\bulletNektar\_BC\_PEC - ∙\\bulletNektar\_UseBC\_PEC - ∙\\bulletNektar\_NoBC\_PEC - ∙\\bulletNektar\_BC\_Rotated - ∙\\bulletNektar\_UseBC\_Rotated - ∙\\bulletNektar\_NoBC\_Rotated
- ∙\\bulletNektar\_InitialConditions - ∙\\bulletNektar\_FunctionShape - ∙\\bulletNektar\_Func\_Expression - ∙\\bulletNektar\_UseFunc\_Expression - ∙\\bulletNektar\_NoFunc\_Expression - ∙\\bulletNektar\_Func\_StaticFile - ∙\\bulletNektar\_UseFunc\_StaticFile - ∙\\bulletNektar\_NoFunc\_StaticFile - ∙\\bulletNektar\_Func\_TransientFile - ∙\\bulletNektar\_UseFunc\_TransientFile - ∙\\bulletNektar\_NoFunc\_TransientFile - ∙\\bulletNektar\_FunctionName - ∙\\bulletNektar\_FN\_InitialConditions - ∙\\bulletNektar\_UseFN\_InitialConditions - ∙\\bulletNektar\_NoFN\_InitialConditions - ∙\\bulletNektar\_FN\_ExactSolution - ∙\\bulletNektar\_UseFN\_ExactSolution - ∙\\bulletNektar\_NoFN\_ExactSolution - ∙\\bulletNektar\_FN\_Forcing - ∙\\bulletNektar\_UseFN\_Forcing - ∙\\bulletNektar\_NoFN\_Forcing - ∙\\bulletNektar\_FN\_AdvectionVelocity - ∙\\bulletNektar\_UseFN\_AdvectionVelocity - ∙\\bulletNektar\_NoFN\_AdvectionVelocity - ∙\\bulletNektar\_FN\_BaseFlow - ∙\\bulletNektar\_UseFN\_BaseFlow - ∙\\bulletNektar\_NoFN\_BaseFlow - ∙\\bulletNektar\_FN\_VCSFields - ∙\\bulletNektar\_UseFN\_VCSFields - ∙\\bulletNektar\_NoFN\_VCSFields - ∙\\bulletNektar\_FN\_MovingReferenceFrame - ∙\\bulletNektar\_UseFN\_MovingReferenceFrame - ∙\\bulletNektar\_NoFN\_MovingReferenceFrame - ∙\\bulletNektar\_FN\_BodyForce - ∙\\bulletNektar\_UseFN\_BodyForce - ∙\\bulletNektar\_NoFN\_BodyForce - ∙\\bulletNektar\_FN\_A\_0 - ∙\\bulletNektar\_UseFN\_A\_0 - ∙\\bulletNektar\_NoFN\_A\_0 - ∙\\bulletNektar\_FN\_MaterialProperties - ∙\\bulletNektar\_UseFN\_MaterialProperties - ∙\\bulletNektar\_NoFN\_MaterialProperties - ∙\\bulletNektar\_FN\_WaterDepth - ∙\\bulletNektar\_UseFN\_WaterDepth - ∙\\bulletNektar\_NoFN\_WaterDepth - ∙\\bulletNektar\_FN\_Coriolis - ∙\\bulletNektar\_UseFN\_Coriolis - ∙\\bulletNektar\_NoFN\_Coriolis - ∙\\bulletNektar\_FN\_DiffusionCoefficient - ∙\\bulletNektar\_UseFN\_DiffusionCoefficient - ∙\\bulletNektar\_NoFN\_DiffusionCoefficient - ∙\\bulletNektar\_FN\_FlowrateForce - ∙\\bulletNektar\_UseFN\_FlowrateForce - ∙\\bulletNektar\_NoFN\_FlowrateForce - ∙\\bulletNektar\_FN\_SpongeCoefficient - ∙\\bulletNektar\_UseFN\_SpongeCoefficient - ∙\\bulletNektar\_NoFN\_SpongeCoefficient - ∙\\bulletNektar\_FN\_Source - ∙\\bulletNektar\_UseFN\_Source - ∙\\bulletNektar\_NoFN\_Source
- ∙\\bulletNektar\_Driver - ∙\\bulletNektar\_DriverType - ∙\\bulletNektar\_Driver\_Standard - ∙\\bulletNektar\_Driver\_ModifiedArnoldi - ∙\\bulletNektar\_Driver\_Arpack - ∙\\bulletNektar\_Driver\_SteadyState - ∙\\bulletNektar\_Driver\_Adaptive - ∙\\bulletNektar\_Driver\_Parareal - ∙\\bulletNektar\_Driver\_PFASST - ∙\\bulletNektar\_EvolutionOperator - ∙\\bulletNektar\_EvOp\_Nonlinear - ∙\\bulletNektar\_EvOp\_Direct - ∙\\bulletNektar\_EvOp\_Adjoint - ∙\\bulletNektar\_EvOp\_TransientGrowth - ∙\\bulletNektar\_EvOp\_SkewSymmetric - ∙\\bulletNektar\_EvOp\_AdaptiveSFD
- ∙\\bulletNektar\_Filters - ∙\\bulletNektar\_Filter\_HistoryPoints - ∙\\bulletNektar\_UseFilter\_HistoryPoints - ∙\\bulletNektar\_NoFilter\_HistoryPoints - ∙\\bulletNektar\_Filter\_Error - ∙\\bulletNektar\_UseFilter\_Error - ∙\\bulletNektar\_NoFilter\_Error - ∙\\bulletNektar\_Filter\_ModalEnergy - ∙\\bulletNektar\_UseFilter\_ModalEnergy - ∙\\bulletNektar\_NoFilter\_ModalEnergy - ∙\\bulletNektar\_Filter\_AeroForces - ∙\\bulletNektar\_UseFilter\_AeroForces - ∙\\bulletNektar\_NoFilter\_AeroForces - ∙\\bulletNektar\_Filter\_AverageFields - ∙\\bulletNektar\_UseFilter\_AverageFields - ∙\\bulletNektar\_NoFilter\_AverageFields - ∙\\bulletNektar\_Filter\_LagrangianPoints - ∙\\bulletNektar\_UseFilter\_LagrangianPoints - ∙\\bulletNektar\_NoFilter\_LagrangianPoints - ∙\\bulletNektar\_Filter\_ReynoldsStresses - ∙\\bulletNektar\_UseFilter\_ReynoldsStresses - ∙\\bulletNektar\_NoFilter\_ReynoldsStresses - ∙\\bulletNektar\_Filter\_AeroForcesSPM - ∙\\bulletNektar\_UseFilter\_AeroForcesSPM - ∙\\bulletNektar\_NoFilter\_AeroForcesSPM - ∙\\bulletNektar\_Filter\_Checkpoint - ∙\\bulletNektar\_UseFilter\_Checkpoint - ∙\\bulletNektar\_NoFilter\_Checkpoint - ∙\\bulletNektar\_Filter\_Energy - ∙\\bulletNektar\_UseFilter\_Energy - ∙\\bulletNektar\_NoFilter\_Energy - ∙\\bulletNektar\_Filter\_BodyFittedVelocity - ∙\\bulletNektar\_UseFilter\_BodyFittedVelocity - ∙\\bulletNektar\_NoFilter\_BodyFittedVelocity - ∙\\bulletNektar\_Filter\_FieldConvert - ∙\\bulletNektar\_UseFilter\_FieldConvert - ∙\\bulletNektar\_NoFilter\_FieldConvert - ∙\\bulletNektar\_Filter\_MaxMinFields - ∙\\bulletNektar\_UseFilter\_MaxMinFields - ∙\\bulletNektar\_NoFilter\_MaxMinFields - ∙\\bulletNektar\_Filter\_Python - ∙\\bulletNektar\_UseFilter\_Python - ∙\\bulletNektar\_NoFilter\_Python
- ∙\\bulletNektar\_Stabilisation - ∙\\bulletNektar\_SpectralVanishingViscosity - ∙\\bulletNektar\_SVV\_Off - ∙\\bulletNektar\_SVV\_ExpKernel - ∙\\bulletNektar\_SVV\_DGKernel - ∙\\bulletNektar\_Dealiasing - ∙\\bulletNektar\_Dealias\_Homogeneous - ∙\\bulletNektar\_UseDealias\_Homogeneous - ∙\\bulletNektar\_NoDealias\_Homogeneous - ∙\\bulletNektar\_Dealias\_SpectralHP - ∙\\bulletNektar\_UseDealias\_SpectralHP - ∙\\bulletNektar\_NoDealias\_SpectralHP - ∙\\bulletNektar\_Dealias\_USEFFT - ∙\\bulletNektar\_UseDealias\_USEFFT - ∙\\bulletNektar\_NoDealias\_USEFFT - ∙\\bulletNektar\_ShockCapture - ∙\\bulletNektar\_ShockCapture\_Off - ∙\\bulletNektar\_ShockCapture\_NonSmooth - ∙\\bulletNektar\_ShockCapture\_Physical - ∙\\bulletNektar\_GJPStabilisation - ∙\\bulletNektar\_GJP\_Off - ∙\\bulletNektar\_GJP\_Explicit - ∙\\bulletNektar\_GJP\_SemiImplicit - ∙\\bulletNektar\_GJP\_Implicit
- ∙\\bulletNektar\_Homogeneous - ∙\\bulletNektar\_HomogeneousDimension - ∙\\bulletNektar\_HomogeneousOff - ∙\\bulletNektar\_Homogeneous1D - ∙\\bulletNektar\_Homogeneous2D - ∙\\bulletNektar\_ModeType - ∙\\bulletNektar\_MultipleModes - ∙\\bulletNektar\_SingleMode - ∙\\bulletNektar\_HalfMode - ∙\\bulletNektar\_ModeType\_NA
- ∙\\bulletNektar\_Forcing - ∙\\bulletNektar\_Force\_Body - ∙\\bulletNektar\_UseForce\_Body - ∙\\bulletNektar\_NoForce\_Body - ∙\\bulletNektar\_Force\_Absorption - ∙\\bulletNektar\_UseForce\_Absorption - ∙\\bulletNektar\_NoForce\_Absorption - ∙\\bulletNektar\_Force\_MovingReferenceFrame - ∙\\bulletNektar\_UseForce\_MovingReferenceFrame - ∙\\bulletNektar\_NoForce\_MovingReferenceFrame

### C\.3Solvers — Spectral Julia

- ∙\\bulletSP: Transform Type - ∙\\bulletSP: Fourier \(FFT\) - ∙\\bulletSP: Chebyshev - ∙\\bulletSP: Mixed Fourier\-Chebyshev
- ∙\\bulletSP: Time Integration - ∙\\bulletSP: Time Scheme - ∙\\bulletSP: ETDRK4 - ∙\\bulletSP: IMEX\-RK - ∙\\bulletSP: Strang Splitting - ∙\\bulletSP: Explicit RK4 - ∙\\bulletSP: Custom Integrator - ∙\\bulletSP: Time Step - ∙\\bulletSP: Fixed dt - ∙\\bulletSP: Adaptive \(error\-based\) - ∙\\bulletSP: Fixed CFL
- ∙\\bulletSP: Resolution - ∙\\bulletSP: N=64\-128 - ∙\\bulletSP: N=256\-512 - ∙\\bulletSP: N=1024\+ - ∙\\bulletSP: Custom N
- ∙\\bulletSP: Operator Splitting - ∙\\bulletSP: SplitODE \(Linear \+ Nonlinear\) - ∙\\bulletSP: Full Explicit \(no split\) - ∙\\bulletSP: Full Implicit - ∙\\bulletSP: Custom Split
- ∙\\bulletSP: Dimensionality - ∙\\bulletSP: 1D - ∙\\bulletSP: 2D - ∙\\bulletSP: 3D - ∙\\bulletSP: Custom Dimensionality
- ∙\\bulletSP: Domain BC - ∙\\bulletSP: Fully Periodic - ∙\\bulletSP: Wall Bounded All - ∙\\bulletSP: Mixed Periodic Wall - ∙\\bulletSP: Custom BC
- ∙\\bulletSP: FFT Backend - ∙\\bulletSP: FFTW CPU - ∙\\bulletSP: CUFFT GPU - ∙\\bulletSP: AbstractFFTs - ∙\\bulletSP: Custom FFT Backend
- ∙\\bulletSP: Implicit Linear Solver - ∙\\bulletSP: Diagonal in Fourier - ∙\\bulletSP: Banded Chebyshev Shen - ∙\\bulletSP: Tau Method - ∙\\bulletSP: Iterative CG GMRES - ∙\\bulletSP: Not Applicable - ∙\\bulletSP: Custom Implicit Solver

### C\.4Solvers — Trixi\.jl

- ∙\\bulletTrixi\_AMR\_ShockCapturing - ∙\\bulletTrixi\_AMR\_Controllers - ∙\\bulletTrixi\_NoAMR - ∙\\bulletTrixi\_ControllerThreeLevel - ∙\\bulletTrixi\_CustomAMRController - ∙\\bulletTrixi\_AMR\_Indicators - ∙\\bulletTrixi\_AMR\_NoIndicator - ∙\\bulletTrixi\_AMR\_IndicatorHennemannGassner - ∙\\bulletTrixi\_AMR\_IndicatorLoehner - ∙\\bulletTrixi\_AMR\_IndicatorMax - ∙\\bulletTrixi\_AMR\_CustomAMRIndicator
- ∙\\bulletTrixi\_ShockCapturing - ∙\\bulletTrixi\_ShockCapturingIndicators - ∙\\bulletTrixi\_NoShockCapturingIndicator - ∙\\bulletTrixi\_IndicatorHennemannGassner - ∙\\bulletTrixi\_CustomShockCapturingIndicator - ∙\\bulletTrixi\_Limiters - ∙\\bulletTrixi\_NoSubcellLimiter - ∙\\bulletTrixi\_SubcellLimiterIDP - ∙\\bulletTrixi\_CustomSubcellLimiter
- ∙\\bulletTrixi\_Callbacks - ∙\\bulletTrixi\_StepCallbacks - ∙\\bulletTrixi\_AMRCallback - ∙\\bulletTrixi\_UseAMRCallback - ∙\\bulletTrixi\_NoAMRCallback - ∙\\bulletTrixi\_AliveCallback - ∙\\bulletTrixi\_UseAliveCallback - ∙\\bulletTrixi\_NoAliveCallback - ∙\\bulletTrixi\_AnalysisCallback - ∙\\bulletTrixi\_UseAnalysisCallback - ∙\\bulletTrixi\_NoAnalysisCallback - ∙\\bulletTrixi\_GlmSpeedCallback - ∙\\bulletTrixi\_UseGlmSpeedCallback - ∙\\bulletTrixi\_NoGlmSpeedCallback - ∙\\bulletTrixi\_LBMCollisionCallback - ∙\\bulletTrixi\_UseLBMCollisionCallback - ∙\\bulletTrixi\_NoLBMCollisionCallback - ∙\\bulletTrixi\_SaveRestartCallback - ∙\\bulletTrixi\_UseSaveRestartCallback - ∙\\bulletTrixi\_NoSaveRestartCallback - ∙\\bulletTrixi\_SaveSolutionCallback - ∙\\bulletTrixi\_UseSaveSolutionCallback - ∙\\bulletTrixi\_NoSaveSolutionCallback - ∙\\bulletTrixi\_StepsizeCallback - ∙\\bulletTrixi\_UseStepsizeCallback - ∙\\bulletTrixi\_CFL - ∙\\bulletTrixi\_CFL\_0\_point\_3 - ∙\\bulletTrixi\_CFL\_0\_point\_5 - ∙\\bulletTrixi\_CFL\_0\_point\_8 - ∙\\bulletTrixi\_CFL\_1\_point\_0 - ∙\\bulletTrixi\_CFL\_1\_point\_25 - ∙\\bulletTrixi\_CFL\_Custom - ∙\\bulletTrixi\_NoStepsizeCallback - ∙\\bulletTrixi\_SummaryCallback - ∙\\bulletTrixi\_UseSummaryCallback - ∙\\bulletTrixi\_NoSummaryCallback - ∙\\bulletTrixi\_VisualizationCallback - ∙\\bulletTrixi\_UseVisualizationCallback - ∙\\bulletTrixi\_NoVisualizationCallback - ∙\\bulletTrixi\_StageCallbacks - ∙\\bulletTrixi\_PositivityPreservingLimiterZhangShu - ∙\\bulletTrixi\_UsePositivityPreservingLimiterZhangShu - ∙\\bulletTrixi\_NoPositivityPreservingLimiterZhangShu - ∙\\bulletTrixi\_SubcellLimiterIDPCorrection - ∙\\bulletTrixi\_UseSubcellLimiterIDPCorrection - ∙\\bulletTrixi\_NoSubcellLimiterIDPCorrection
- ∙\\bulletTrixi\_IC\_BC - ∙\\bulletTrixi\_BoundaryConditions - ∙\\bulletTrixi\_BoundaryConditionDirichlet - ∙\\bulletTrixi\_UseBoundaryConditionDirichlet - ∙\\bulletTrixi\_NoBoundaryConditionDirichlet - ∙\\bulletTrixi\_BoundaryConditionNavierStokesWall - ∙\\bulletTrixi\_UseBoundaryConditionNavierStokesWall - ∙\\bulletTrixi\_NoBoundaryConditionNavierStokesWall - ∙\\bulletTrixi\_BoundaryConditionSlipWall - ∙\\bulletTrixi\_UseBoundaryConditionSlipWall - ∙\\bulletTrixi\_NoBoundaryConditionSlipWall - ∙\\bulletTrixi\_PeriodicBC - ∙\\bulletTrixi\_UsePeriodicBC - ∙\\bulletTrixi\_NoPeriodicBC - ∙\\bulletTrixi\_BoundaryConditionSupersonicOutflow - ∙\\bulletTrixi\_UseBoundaryConditionSupersonicOutflow - ∙\\bulletTrixi\_NoBoundaryConditionSupersonicOutflow - ∙\\bulletTrixi\_BoundaryConditionMixed - ∙\\bulletTrixi\_UseBoundaryConditionMixed - ∙\\bulletTrixi\_NoBoundaryConditionMixed
- ∙\\bulletTrixi\_Solvers - ∙\\bulletTrixi\_DGMulti - ∙\\bulletTrixi\_DGSEM - ∙\\bulletTrixi\_polydeg - ∙\\bulletTrixi\_polydeg\_2 - ∙\\bulletTrixi\_polydeg\_3 - ∙\\bulletTrixi\_polydeg\_4 - ∙\\bulletTrixi\_polydeg\_5 - ∙\\bulletTrixi\_polydeg\_Custom - ∙\\bulletTrixi\_FDSBP - ∙\\bulletTrixi\_FV
- ∙\\bulletTrixi\_TimeIntegration - ∙\\bulletTrixi\_CarpenterKennedy2N - ∙\\bulletTrixi\_CarpenterKennedy2N\_Variant - ∙\\bulletTrixi\_CarpenterKennedy2N54 - ∙\\bulletTrixi\_CarpenterKennedy2N43 - ∙\\bulletTrixi\_OrdinaryDiffEq - ∙\\bulletTrixi\_OrdinaryDiffEq\_Algorithm - ∙\\bulletTrixi\_OrdinaryDiffEq\_SSPRK43 - ∙\\bulletTrixi\_OrdinaryDiffEq\_SSPRK33 - ∙\\bulletTrixi\_OrdinaryDiffEq\_Tsit5 - ∙\\bulletTrixi\_OrdinaryDiffEq\_RK4 - ∙\\bulletTrixi\_OrdinaryDiffEq\_Rosenbrock23 - ∙\\bulletTrixi\_OrdinaryDiffEq\_RDPK3SpFSAL49 - ∙\\bulletTrixi\_OrdinaryDiffEq\_CKLLSRK54\_3C - ∙\\bulletTrixi\_OrdinaryDiffEq\_Vern7 - ∙\\bulletTrixi\_OrdinaryDiffEq\_Vern9 - ∙\\bulletTrixi\_OrdinaryDiffEq\_Custom - ∙\\bulletTrixi\_PairedExplicitRK - ∙\\bulletTrixi\_ParsaniKetcheson3Sstar - ∙\\bulletTrixi\_SimpleSSPRK33
- ∙\\bulletTrixi\_Meshes - ∙\\bulletTrixi\_DGMultiMesh - ∙\\bulletTrixi\_P4estMesh - ∙\\bulletTrixi\_StructuredMesh - ∙\\bulletTrixi\_T8codeMesh - ∙\\bulletTrixi\_TreeMesh - ∙\\bulletTrixi\_UnstructuredMesh2D
- ∙\\bulletTrixi\_Equations - ∙\\bulletTrixi\_HyperbolicEquation - ∙\\bulletTrixi\_AcousticPerturbation - ∙\\bulletTrixi\_Burgers - ∙\\bulletTrixi\_CompressibleEuler - ∙\\bulletTrixi\_CompressibleEulerMulticomponent - ∙\\bulletTrixi\_HyperbolicDiffusion - ∙\\bulletTrixi\_IdealGlmMhd - ∙\\bulletTrixi\_LatticeBoltzmann - ∙\\bulletTrixi\_LinearAdvection - ∙\\bulletTrixi\_LinearElasticity - ∙\\bulletTrixi\_LinearizedEuler - ∙\\bulletTrixi\_Maxwell - ∙\\bulletTrixi\_NonidealEuler - ∙\\bulletTrixi\_PassiveTracers - ∙\\bulletTrixi\_PolytropicEuler - ∙\\bulletTrixi\_TrafficFlow - ∙\\bulletTrixi\_CustomHyperbolic - ∙\\bulletTrixi\_ParabolicEquation - ∙\\bulletTrixi\_NoParabolic - ∙\\bulletTrixi\_CompressibleNavierStokes - ∙\\bulletTrixi\_LaplaceDiffusion - ∙\\bulletTrixi\_LinearDiffusion - ∙\\bulletTrixi\_CustomParabolic - ∙\\bulletTrixi\_SourceTerm - ∙\\bulletTrixi\_NoSource - ∙\\bulletTrixi\_BodyForce - ∙\\bulletTrixi\_ReactionSource - ∙\\bulletTrixi\_ForcingMMS - ∙\\bulletTrixi\_CustomSource
- ∙\\bulletTrixi\_NumericalFluxes - ∙\\bulletTrixi\_VolumeFluxDG - ∙\\bulletTrixi\_VolumeFluxDG\_Ranocha - ∙\\bulletTrixi\_VolumeFluxDG\_Chandrashekar - ∙\\bulletTrixi\_VolumeFluxDG\_KennedyGruber - ∙\\bulletTrixi\_VolumeFluxDG\_ShimaEtal - ∙\\bulletTrixi\_VolumeFluxDG\_Central - ∙\\bulletTrixi\_VolumeFluxDG\_Custom - ∙\\bulletTrixi\_VolumeFluxFV - ∙\\bulletTrixi\_VolumeFluxFV\_LaxFriedrichs\_naive - ∙\\bulletTrixi\_VolumeFluxFV\_LaxFriedrichs\_default - ∙\\bulletTrixi\_VolumeFluxFV\_HLL - ∙\\bulletTrixi\_VolumeFluxFV\_HLLC - ∙\\bulletTrixi\_VolumeFluxFV\_Custom - ∙\\bulletTrixi\_SurfaceFlux - ∙\\bulletTrixi\_SurfaceFlux\_LaxFriedrichs\_naive - ∙\\bulletTrixi\_SurfaceFlux\_LaxFriedrichs\_default - ∙\\bulletTrixi\_SurfaceFlux\_HLL - ∙\\bulletTrixi\_SurfaceFlux\_HLLC - ∙\\bulletTrixi\_SurfaceFlux\_Custom
- ∙\\bulletTrixi\_ecosystem