从一维基准测试到复杂三维几何结构：智能体能否可靠地为自动化偏微分方程仿真定义问题？

# From 1D Benchmarks to Complex 3D Geometries: Can Agents Reliably Specify Problems for Automated PDE Simulation?
Source: [https://arxiv.org/html/2610.41267](https://arxiv.org/html/2610.41267)
Akshat Jani Sakhinana Sagar Srinivas Venkataramana RunkanaTata Research Development and Design Center, Pune, India 411057††thanks:Correspondence to: Sakhinana Sagar Srinivas \([sagar\.sakhinana@tcs\.com](mailto:[email protected])\)\.

###### Abstract

Turning an engineering request into a simulation requires consistent geometry, physics, boundary conditions, and numerical settings\. Recent agent\-assisted systems automate simulation setup, but successful execution alone does not establish whether the specification preserves the intended problem or correctly handles missing information\. We study whether agents can reliably construct complete executable JSON specifications for automated geometry creation, meshing, PDE solving, and result export\. Our workflow combines model extraction with deterministic scientific and capability checks, explicit clarification, and validated local patches that preserve unrelated accepted fields\. We evaluate specification correctness, execution completion, and numerical verification separately\. On 120 template\-related requests, the full workflow achieves 98\.3% correct first\-turn specification and decision outcomes, compared with 92\.5% for schema\-assisted single shot and 82\.5% for model\-only: gains of 5\.8 and 15\.8 percentage points, respectively\. It makes no false\-ready decisions among the 42 requests requiring clarification or rejection\. Component ablations identify contributions from scientific checks and conditional repair\. Separate downstream studies cover 1D, 2D, and 3D steady and unsteady, linear and nonlinear problems, including flow and elasticity with complex three\-dimensional wing geometries\. A 30\-case gallery demonstrates automated execution from retained JSON specifications, while 9 of 12 selected live user\-to\-results tests achieve joint specification, execution, export, and numerical\-reference success \(75\.0%\)\. These results support traceable, engineer\-reviewed simulation assistance within the tested scope, while fresh\-response regressions identify consistency as a priority for further improvement\.

## 1Introduction

An engineer designing a battery cooling system, a water pump, or a drone wing faces a common task: turn a physical question into a simulation that can support a design decision\. This requires consistent geometry, governing equations, material properties, boundary conditions, mesh, and solver settings\. Boundary conditions must refer to the correct surfaces, and numerical settings must match the equation and time regime\. Three\-dimensional geometries, nonlinear equations, and time\-dependent conditions make this coordination more demanding\.

Language\-model agents can assist by interpreting requests, collecting missing information, preparing configurations, and coordinating simulation tools\. ReAct demonstrates how reasoning can be combined with actions and external feedback\([Yao et al\., 2023](https://arxiv.org/html/2610.41267#bib.bib8)\)\. The aim is to accelerate routine setup and iteration so that engineers can focus on unresolved physical questions, analysis, and design decisions\. Engineers retain responsibility for choosing appropriate models and assessing the results\. Useful assistance therefore requires preserving the intended physics and keeping unresolved information visible\.

Recent systems demonstrate progress in simulation automation\. MetaOpenFOAM uses collaborating agents and retrieved examples to configure and execute OpenFOAM cases\([Chen et al\., 2024](https://arxiv.org/html/2610.41267#bib.bib1)\), while Foam\-Agent 2\.0 covers meshing, execution, and visualization\([Yue et al\., 2025](https://arxiv.org/html/2610.41267#bib.bib2)\)\. ChatCFD combines domain knowledge with error diagnosis and evaluates physical fidelity beyond successful execution\([Fan et al\., 2026](https://arxiv.org/html/2610.41267#bib.bib3)\)\. For finite\-element simulation, MechAgents explores collaborating agents for elasticity problems\([Ni and Buehler, 2024](https://arxiv.org/html/2610.41267#bib.bib5)\), and ALL\-FEM combines specialized agents with fine\-tuned models for FEniCS code generation\([Deotale et al\., 2026](https://arxiv.org/html/2610.41267#bib.bib6)\)\. JSONSchemaBench examines structured\-output generation through schema coverage, efficiency, and output quality\([Geng et al\., 2025](https://arxiv.org/html/2610.41267#bib.bib7)\)\. Together, these studies motivate examining whether an accepted specification represents the request, contains sufficient information, and remains consistent after corrections\.

PDEFlow directly motivates this study\([Jani et al\., 2026](https://arxiv.org/html/2610.41267#bib.bib4)\)\. It connects conversational specification with finite\-element data generation, neural\-operator training, and checkpoint\-based inference\. Its stateful workflow supports local updates, with a formulation covering linear one\- and two\-dimensional problems\. An incorrect specification can affect subsequent simulations, training data, and predictions\. Here, we examine specification reliability while extending the supported simulation scope to nonlinear equations and complex three\-dimensional geometries\.

We therefore ask:

> Can agents reliably construct complete JSON specifications that preserve the intended physics, identify missing or unsupported information, and trigger automated geometry creation, meshing, solving, and result export?

Our study builds on these advances by examining the reliability of the scientific specification and the decision to permit execution\. We address three connected research gaps\.

First, successful execution alone cannot establish whether a system correctly handles incomplete, conflicting, or unsupported requests\. We test whether it should produce a complete specification, ask for clarification, or reject the request\. Scientific fields are checked against fixed references, and incorrect decisions to proceed are measured separately\. Clarification completion is evaluated separately from the initial decision to ask a question\.

Second, repairing a configuration must preserve the intended problem beyond the corrected field\. We use a retained JSON specification as the interface between the agent and reviewed numerical routines\. The model proposes content; deterministic code checks its structure, request evidence, and supported capabilities\. Clarifications and repairs use local patches restricted to authorized paths, with revision checks and validation before acceptance\. This makes changes traceable and protects unrelated accepted fields\. Component ablations examine how these checks and repair mechanisms affect correctness\.

Third, specification correctness, execution completion, and numerical accuracy require different evidence\. We evaluate user\-to\-JSON outcomes, numerical verification, and execution from retained specifications separately\. Reference comparisons and convergence studies assess numerical accuracy; the gallery assesses downstream execution; selected live tests assess the complete user\-to\-results path\. These studies retain distinct denominators and cover supported 1D/2D/3D, steady/unsteady, and linear/nonlinear benchmarks, including flow and elasticity with three\-dimensional wing geometries\.

This design provides a practical basis for production integration\. An explicit JSON interface connects agents to engineering software while keeping assumptions and accepted changes available for review\. Reviewed solver templates, controlled updates, and retained records support repeatable operation\. Commercial deployment would additionally require application\-specific validation, access and computing\-budget controls, and engineer approval for consequential decisions\. Our study evaluates the underlying reliability mechanisms and identifies where further development is needed\.

## 2Methodology

The framework separates specification construction from numerical execution \(Figure[1](https://arxiv.org/html/2610.41267#S2.F1)\)\. A language model interprets the request and proposes a JSON specification\. Deterministic code checks the specification, manages authorized changes, and controls whether it can enter the numerical workflow\.

![Refer to caption](https://arxiv.org/html/2610.41267v1/results/overall_system_diagram.png)Figure 1:Overall workflow, from left to right\. \(a\) User/frontend: receives PDE requests and clarification replies\. \(b\) Agentic and validation system: constructs and checks the specification while retaining session history\. \(c\) Validated JSON: records the problem and numerical settings\. \(d\) Execution backend: Gmsh creates geometry and meshes, FEniCSx and PETSc solve the PDEs, and ParaView supports visualization\. \(e\) Outputs: returns fields, plots, diagnostics, and downloadable artifacts\. The feedback loop supports validated user revisions\.### 2\.1The executable JSON specification

The JSON specification describes the physical problem and the numerical settings required to run it \(Table[1](https://arxiv.org/html/2610.41267#S2.T1)\)\. Fields follow explicit types: a scalar value is represented as a constant or an expression, and a vector contains scalar components\. Boundary conditions identify both their target boundaries and the required values\. This structure preserves spatially varying expressions rather than reducing them to constants\.

Table 1:Main fields of the executable JSON specification\. The original request, schema version, and descriptive labels are also retained\.The initial draft also records ambiguities and the source of extracted values\. Provenance records link values to quoted request text and its character positions, or identify a permitted default policy\. Required information that is missing remains unresolved\. Defaults are distinguishable from user\-supplied facts, and a partial draft becomes executable only after it satisfies the complete problem contract\.

### 2\.2Constructing the specification

The initial model call receives the user request, extraction instructions, and a strict JSON schema\. A vocabulary\-based router narrows the geometry and equation choices when it recognizes the request; otherwise, the full set of relevant choices remains available\. Reviewed case aliases, when present, help identify supported problem families without supplying unrequested physical values or benchmark answers\. The same model is used for extraction and any subsequent model repair\.

The response is parsed into a retained draft\. Deterministic normalization handles supported equivalent representations, such as alternative enum spellings and evidence\-path notation\. Evidence positions are checked against the unchanged request, and raw responses and normalization records are retained for inspection\. Schema\-constrained generation controls the response structure; the checks in Figure[2](https://arxiv.org/html/2610.41267#S2.F2)assess its scientific consistency and suitability for execution\.

![Refer to caption](https://arxiv.org/html/2610.41267v1/results/agentic_system_diagram.png)Figure 2:Specification construction and validation\. The model proposes a draft or local repair, while deterministic checks assess structure, request evidence, and solver compatibility\. Missing information returns to the user for clarification\. Session storage retains the specification, revisions, and audit records\. Execution requires a complete specification with no blocking findings\.
### 2\.3Validation, clarification, and patch repair

Validation checks field types, relationships between sections, agreement with request evidence, and compatibility with supported numerical capabilities\. These checks cover dimensions, mesh and boundary conventions, scalar/vector fields, time settings, and solver requirements\. The decision isready,needs\_clarification, orrejected\. Readiness requires a complete specification with no blocking findings\. Missing information produces a clarification question, ordered by field dependencies and a fixed priority\. Unsupported requests and explicit contradictions are rejected\. These checks have finite coverage and do not prove scientific correctness\.

Clarifications, amendments, and repairs update the retained specification through versioned add, replace, or remove operations tied to a source turn and base revision\. Application code determines the permitted paths; changing an accepted value requires user authorization\. A model repair receives the findings, permitted paths, and nearby context, with at most one attempt per accepted user turn\. Fully determined boundary corrections can instead be constructed directly from independently matched request data\. Each proposal is applied to a private copy and checked for revision, scope, and validity before acceptance\. Invalid or stale proposals leave the accepted specification unchanged\. The original request, prior evidence, and patch audits are retained\. Execution permission is recomputed after updates and session reloads\. Appendix[A](https://arxiv.org/html/2610.41267#A1)gives the detailed contract\.

### 2\.4Geometry, mesh generation, and PDE solving

The executor maps the validated specification to a registered solver template\. Reviewed code defines the geometry builders and finite\-element weak forms; the model does not generate executable solver code\. Structured builders or Gmsh create the mesh, and boundary selectors identify the corresponding mesh facets\. Complex geometries distinguish structural solids from external\-flow domains and retain their physical boundary groups\. Mathematical expressions are evaluated through a restricted parser\. FEniCSx/UFL constructs and assembles the finite\-element forms, while PETSc solves the linear or nonlinear systems\.

Scalar templates cover supported 1D/2D/3D problems; linear elasticity and steady incompressible flow cover supported 2D/3D problems\. Transient heat and nonlinear reaction\-diffusion use backward Euler or Crank–Nicolson time integration\. Geometry and equation combinations are checked explicitly rather than assumed to be interchangeable\. Each result bundle retains the accepted specification, mesh and field exports, solver diagnostics, and execution status\. Reference comparisons and convergence studies assess numerical accuracy separately \(Appendix[B](https://arxiv.org/html/2610.41267#A2)\)\. Specification acceptance, execution completion, and numerical verification therefore remain distinct outcomes\.

## 3Experiments and Results

### 3\.1Experimental setup

We evaluate three stages separately: constructing JSON from user requests, verifying numerical solutions, and executing retained JSON specifications\. Tables[2](https://arxiv.org/html/2610.41267#S3.T2)–[4](https://arxiv.org/html/2610.41267#S3.T4)summarize the studies\. Paper IDs identify experiments and gallery groups; their original records are mapped in Appendix[E](https://arxiv.org/html/2610.41267#A5)\.

#### Agentic studies: user requests to JSON\.

We use gpt\-5\.4\-mini\-2026\-03\-17 with reasoning effort set tonone, a 6,400\-token output limit, and workflow v1\.8 with at most one model patch repair\. The main benchmark contains 120 requests: 48 extraction, 30 complete\-specification, 30 clarification, and 12 rejection/conflict tasks\. Half use natural wording and half use schema\-explicit wording\. The requests are code\-authored and share related templates\. Scalar problems cover supported 1D/2D/3D, steady/unsteady, and linear/nonlinear settings\.

Table 2:Agentic studies: user requests to JSON\. Studies reuse requests, so their counts are not added together\.
#### Numerical verification\.

We compare solutions with analytical or manufactured references using integrated finite\-element errors and predefined acceptance rules\. Scalar and elasticity studies use P1 elements; incompressible flow uses P2/P1 elements\. Structured builders or Gmsh generate the meshes, and FEniCSx/PETSc performs the solves\. The latest verification campaign uses the DOLFINx 0\.11\.0 series, Gmsh 4\.12\.1, and PETSc 3\.19\.6\. Exact build versions and hardware are recorded in Appendix[E](https://arxiv.org/html/2610.41267#A5)\. BE and CN denote backward Euler and Crank–Nicolson\.

V6 exceeds the spatial\-isolation limit:10\.84%10\.84\\%compared with the allowed10%10\\%\. The optional finest 3D cantilever study was not attempted\. Complete inputs, refinement sequences, solver settings, and numerical results are provided in Appendix[B](https://arxiv.org/html/2610.41267#A2)\.

Table 3:Numerical verification studies\. Counts include 56 completed study solves, including studies with failed checks\. Development smoke runs are excluded\.IDExperimentVerification setupSolvesV12D elasticityFour mesh levels; displacement and energy checks4V32D steady flowThree mesh levels and one pressure patch test4V42D unsteady heatFive step sizes for each BE/CN scheme; one equilibrium control11V62D annular heatBE/CN refinement and spatial\-isolation check8V72D rectangular heatBE/CN refinement and spatial\-isolation check8V23D elasticityFour mesh levels; displacement and energy checks4V53D nonlinear reaction\-diffusionFive step sizes for each BE/CN scheme; one equilibrium control11V83D parallel agreementSame cantilever problem with 4, 8, 16, and 32 MPI processes4V9Solver tolerance checksTighter tolerances for 2D flow and 3D reaction\-diffusion2Total56
#### Solution gallery: JSON to results\.

We execute 30 retained JSON specifications through geometry creation, meshing, solving, and export: three cases in 1D, sixteen in 2D, and eleven in 3D\. The gallery includes steady and unsteady, linear and nonlinear problems, with flow and elasticity cases involving wing geometries\. Table[4](https://arxiv.org/html/2610.41267#S3.T4)groups the cases by physics and dimension\. Completion establishes execution; numerical qualifications are reported separately in Appendix[F](https://arxiv.org/html/2610.41267#A6)\.

Gallery inputs are retained JSON specifications, so these cases do not measure user\-to\-JSON accuracy\. Separate live user\-to\-results tests evaluate the complete path \(Appendix[C](https://arxiv.org/html/2610.41267#A3)\)\.

Table 4:Solution gallery: JSON to results\. Each case belongs to one group\.IDExperiment groupScopeCasesG1Rod and polyline diffusion1D; steady/unsteady; linear3G2Linear scalar equations2D; steady; linear6G3Nonlinear diffusion2D; steady; nonlinear2G4Heat and reaction\-diffusion2D; unsteady; linear/nonlinear4G7Airfoil and cylinder flow2D; steady; nonlinear2G9Cantilever and perforated plate2D; steady; linear elasticity2G5Diffusion and conduction3D; steady; linear/nonlinear5G6Heat and reaction\-diffusion3D; unsteady; linear/nonlinear2G8Wing and sphere flow3D; steady; nonlinear2G10Wing and solid cantilevers3D; steady; linear elasticity2Total30

### 3\.2Tasks, baselines, and evaluation measures

#### Agentic studies \(A1–A3\)\.

A1 measures whether the first response contains the expected scientific fields and correctly decides to proceed, request clarification, or reject\. All systems use the same fixed references and scorer\. The 120 requests include 78 complete requests and 42 requiring clarification or rejection\. We report first\-turn correctness over all 120 requests, complete\-specification correctness over the 78 complete requests, and false\-ready decisions over the 42 non\-ready requests\. A false\-ready decision occurs when the system permits execution although clarification or rejection is required\. Valid JSON alone does not establish scientific correctness\.

We compare three systems\. Model\-only produces JSON in one response\. Schema\-assisted single shot adds schema handling and basic deterministic checks\. The full workflow adds request grounding, scientific and capability checks, a readiness decision, and conditional patch repair\. Single shot and the full workflow share the initial model response\. Model\-only uses a different prompt and an independent response, so its comparison also reflects those differences\.

A2 compares each component removal with its own full\-workflow reference\. Deterministic removals reuse saved responses for 120 requests\. Input\-changing controls use new model responses on a fixed 40\-request subset\. We distinguish the number evaluated from the number that actually used the removed component\. A3 applies fixed authorized replies to 30 clarification requests and checks whether a correct complete specification is reached\. This measures completed interaction separately from first\-turn clarification decisions\. API costs and model\-call times are recorded separately from solver time; shared calls are counted once\.

#### Numerical verification \(V1–V9\)\.

V1–V3 test spatial convergence against reference fields, with pressure and energy checks where applicable\. V4–V7 test temporal convergence under BE and CN; V4–V5 include equilibrium controls, and V6–V7 check whether spatial error is sufficiently small to assess temporal accuracy\. We report integratedL2L^\{2\}errors,H1H^\{1\}seminorm errors where applicable, and observed convergence orders\. The spatial\-contamination limit is10%10\\%\. V8 compares parallel solutions with a retained serial solution, using a relative coefficient and quantity\-of\-interest limit of10−710^\{\-7\}\. V9 compares baseline and tighter\-tolerance solutions using a relative field and quantity\-of\-interest limit of10−610^\{\-6\}\. Passing verification requires every declared check; completed solves and unattempted studies are not automatically counted as passes \(Appendix[B](https://arxiv.org/html/2610.41267#A2)\)\.

#### Solution gallery and live workflow tests\.

For G1–G10, execution succeeds when geometry creation, meshing, solving, and the required exports complete\. Numerical accuracy is assessed separately using available references and refinement checks; verified, failed, and preliminary qualifications are retained\. For the 12 selected live user\-to\-results attempts, joint success requires the intended specification, a completed solve, the required artifacts, and a passed numerical reference check\. Blocked attempts, wrong\-intent executions, and reference failures remain in the denominator\. Detailed outcomes appear in Appendices[F](https://arxiv.org/html/2610.41267#A6)and[C](https://arxiv.org/html/2610.41267#A3)\.

### 3\.3Specification construction results

Table[5](https://arxiv.org/html/2610.41267#S3.T5)reports the v1\.8 comparison on all 120 requests in A1\. The full workflow achieves118/120118/120correct first\-turn outcomes \(98\.3%98\.3\\%\), compared with111/120111/120\(92\.5%92\.5\\%\) for schema\-assisted single shot and99/12099/120\(82\.5%82\.5\\%\) for model\-only\. It produces correct complete specifications for76/7876/78complete requests \(97\.4%97\.4\\%\) and makes no false\-ready decisions on the 42 requests requiring clarification or rejection\. In A3, fixed authorized replies lead to correct complete specifications for28/3028/30clarification requests \(93\.3%93\.3\\%\)\. These results describe v1\.8 on the main template\-related benchmark\. Numerical accuracy is assessed separately\. Appendix[D](https://arxiv.org/html/2610.41267#A4)provides detailed task results, clarification studies, and later v1\.9 checks\. The latter distinguish re\-evaluation of saved contexts from construction using fresh model responses\. Replay improvements do not establish a uniform gain in fresh construction or replace the A1 scores\.

Table 5:Specification and decision correctness for A1 \(v1\.8\)\. Denominators are 120 total requests, 78 complete requests, and 42 requests requiring clarification or rejection\. Failed cases remain in the denominators\.
### 3\.4Component ablation results

A2 removes one component at a time from v1\.8 and compares each variant with its full\-workflow reference on the same requests \(Figure[3](https://arxiv.org/html/2610.41267#S3.F3)and Table[6](https://arxiv.org/html/2610.41267#S3.T6)\)\. Deterministic removals reuse saved model responses, and the model\-repair comparison uses recorded outcomes before repair\. Input\-changing controls use fresh model responses on a fixed 40\-request subset\. Component use is reported separately because an evaluated request may never reach the component being removed\.

Removing semantic/capability reconciliation reduces correct outcomes by four, and removing model repair reduces them by three\. Removing scientific grounding or the readiness gate reduces them by one each; mandatory structural checks remain active when grounding is removed\. The output\-format result measures dependence on the required representation\. Mapping hints are not exercised in the 40\-request input\-control subset, so their zero difference does not show that they are unnecessary\. Component effects overlap and must not be added together\.

Table 6:Component ablation results for A2 \(v1\.8\)\. Entries show counts and percentages\. Component exercised counts requests that used the removed component in the reference workflow\. Each block states its full\-workflow reference; all evaluated requests remain in the denominators\.![Refer to caption](https://arxiv.org/html/2610.41267v1/results/component_ablation_combined_scale.png)Figure 3:Component ablation results\. Values are full\-workflow accuracy minus ablated accuracy, in percentage points, on matched requests\. Positive values favour the full workflow\. The 120\-request comparisons use a reference score of118/120118/120; the 40\-request input controls use40/4040/40\. All groups share one horizontal scale\. Output\-format removal causes parsing failures on all 120 requests\. The hollow marker identifies unexercised mapping hints\. Schema and routing controls use new model draws, so their differences also include variation between responses\.
### 3\.5Failure cases and limits of reliability

Failures include omitted or altered expressions, incorrect boundary data, and repairs that do not recover the intended specification\. A correct clarification question does not guarantee a correct completed specification, and readiness checks can miss scientific errors\. Later regressions recover some retained errors but show mixed results when specifications are rebuilt from fresh responses\. Workflow changes and new responses occur together, so the fresh\-response tests do not isolate the cause of each difference\. Related request templates, a fixed model, and the tested versions limit conclusions about unfamiliar engineering requests \(Appendix[D](https://arxiv.org/html/2610.41267#A4)\)\.

Specification correctness, execution completion, and numerical accuracy remain separate requirements\. Gallery cases start from retained JSON and therefore assess downstream execution rather than agent construction\. The selected live tests use simple reference fields, while some numerical studies fail or remain unattempted\. These qualifications are retained in Appendices[B](https://arxiv.org/html/2610.41267#A2),[F](https://arxiv.org/html/2610.41267#A6), and[C](https://arxiv.org/html/2610.41267#A3)\.

### 3\.6Conclusion

We studied whether agents can construct executable JSON specifications for automated geometry creation, meshing, PDE solving, and result export\. On the main benchmark, the full workflow achieves118/120118/120correct first\-turn outcomes \(98\.3%98\.3\\%\), compared with111/120111/120\(92\.5%92\.5\\%\) for schema\-assisted single shot and99/12099/120\(82\.5%82\.5\\%\) for model\-only\. It produces correct complete specifications for76/7876/78complete requests \(97\.4%97\.4\\%\)\. Ablations identify contributions from scientific checks and conditional repair\. Separately, the gallery demonstrates execution from 30 retained JSON specifications across supported steady/unsteady and linear/nonlinear problems, including complex wing geometries\. The v1\.9 live workflow test achieves9/129/12joint successes \(75\.0%75\.0\\%\) on selected known requests\.

These results support simulation assistance within the tested scope, with engineers reviewing assumptions and results\. Broader\-input failures show that general reliability remains unresolved\. Future work should improve preservation of expressions and boundary conditions, strengthen repair coverage, and evaluate independently authored requests with numerical reference checks\.

## References

- Y\. Chen, X\. Zhu, H\. Zhou, and Z\. RenMetaOpenFOAM: an LLM\-based multi\-agent framework for CFD\.arXiv preprint arXiv:2407\.21320\.External Links:[Document](https://dx.doi.org/10.48550/arXiv.2407.21320),[Link](https://arxiv.org/abs/2407.21320)Cited by:[§1](https://arxiv.org/html/2610.41267#S1.p3.1)\.
- Deotaleet al\.\(2026\)R\. Deotale, A\. Srinivasan, M\. Golestanian, Y\. Tian, T\. Zhang, P\. Vlachos, and H\. GomezALL\-FEM: agentic large language models fine\-tuned for finite element methods\.Computer Methods in Applied Mechanics and Engineering457,pp\. 118985\.External Links:[Document](https://dx.doi.org/10.1016/j.cma.2026.118985)Cited by:[§1](https://arxiv.org/html/2610.41267#S1.p3.1)\.
- Fanet al\.\(2026\)E\. Fan, K\. Hu, Z\. Wu, J\. Ge, J\. Miao, Y\. Zhang, H\. Sun, W\. Wang, and T\. ZhangChatCFD: a large language model\-driven agent for end\-to\-end computational fluid dynamics automation with structured knowledge and reasoning\.Advanced Intelligent Discovery2,pp\. e202500174\.External Links:[Document](https://dx.doi.org/10.1002/aidi.202500174),[Link](https://doi.org/10.1002/aidi.202500174)Cited by:[§1](https://arxiv.org/html/2610.41267#S1.p3.1)\.
- Genget al\.\(2025\)S\. Geng, H\. Cooper, M\. Moskal, S\. Jenkins, J\. Berman, N\. Ranchin, R\. West, E\. Horvitz, and H\. NoriJSONSchemaBench: a rigorous benchmark of structured outputs for language models\.External Links:2501\.10868,[Link](https://arxiv.org/abs/2501.10868)Cited by:[§1](https://arxiv.org/html/2610.41267#S1.p3.1)\.
- Janiet al\.\(2026\)A\. Jani, P\. Gadekar, S\. S\. Srinivas, and V\. RunkanaPDEFlow: autonomous agentic PDE pipelines for neural operator learning and solver\-free inference\.arXiv preprint arXiv:2607\.05134\.External Links:[Document](https://dx.doi.org/10.48550/arXiv.2607.05134),[Link](https://arxiv.org/abs/2607.05134)Cited by:[§1](https://arxiv.org/html/2610.41267#S1.p4.1)\.
- Ni and Buehler \(2024\)B\. Ni and M\. J\. BuehlerMechAgents: large language model multi\-agent collaborations can solve mechanics problems, generate new data, and integrate knowledge\.Extreme Mechanics Letters67,pp\. 102131\.External Links:[Document](https://dx.doi.org/10.1016/j.eml.2024.102131)Cited by:[§1](https://arxiv.org/html/2610.41267#S1.p3.1)\.
- Yaoet al\.\(2023\)S\. Yao, J\. Zhao, D\. Yu, N\. Du, I\. Shafran, K\. Narasimhan, and Y\. CaoReAct: synergizing reasoning and acting in language models\.InInternational Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=WE_vluYUL-X)Cited by:[§1](https://arxiv.org/html/2610.41267#S1.p2.1)\.
- Yueet al\.\(2025\)L\. Yue, N\. Somasekharan, T\. Zhang, Y\. Cao, and S\. PanFoam\-Agent 2\.0: an end\-to\-end composable multi\-agent framework for automating CFD simulation in OpenFOAM\.arXiv preprint arXiv:2509\.18178\.External Links:[Document](https://dx.doi.org/10.48550/arXiv.2509.18178),[Link](https://arxiv.org/abs/2509.18178)Cited by:[§1](https://arxiv.org/html/2610.41267#S1.p3.1)\.

## Appendix AJSON Specification, Prompts, and Patch Protocol

The workflow uses three objects: an extraction draft, a complete solver\-facing JSON specification, and a patch that updates the retained draft\. The model proposes content; application code validates it and determines whether execution is permitted\.

### A\.1Complete schema and field requirements

The extraction draft contains “problem”, “ambiguities”, and “provenance”\. Unresolved fields may be null\. Parsing removes null placeholders and applies documented normalization rules before validation\. A draft becomes executable only when it satisfies the complete problem contract and the scientific and capability checks\. Table[7](https://arxiv.org/html/2610.41267#A1.T7)summarizes the solver\-facing fields\. The problem schema uses version 2\.0; workflow and patch versions are recorded separately\.

Table 7:Fields of the solver\-facing JSON specification\. Requirements depend on the selected geometry and equation\.#### Supported representations\.

Geometry branches include intervals and polylines in 1D; rectangles, disks, annuli, L\-shaped and perforated domains, and airfoil flow domains in 2D; and boxes, cylinders, spheres, solid wings, and external\-flow domains in 3D\. Equation branches cover Poisson, Laplace, steady heat, Helmholtz, advection–diffusion, nonlinear Poisson, transient heat, nonlinear reaction–diffusion, linear elasticity, and steady incompressible flow\. Nonlinear branches use the declared power, exponential, or hyperbolic\-sine law\. The capability checks determine which combinations can run; the catalogues do not imply support for every combination\.

Scalars are represented as constants or expressions, and vectors contain scalar components\. Expressions pass through a restricted mathematical parser\. Boundary selectors use named boundaries for structured meshes or registered physical groups for Gmsh meshes\. Dirichlet conditions require a value, Neumann conditions require a flux, and Robin conditions require coefficient and reference fields\. Changing a condition type also requires updating its associated data\.

#### Evidence and defaults\.

Each provenance record identifies the field, extracted value, quoted request fragment, character positions, and origin\. Character positions start at zero and exclude the end position\. The fragment must match the original request exactly, and its field path must resolve to the claimed value\. List\-valued facts are recorded as whole lists\. Generated titles and schema labels are not scientific evidence\.

Documented defaults must identify an applicable policy and remain distinguishable from user\-supplied values\. Missing critical information, such as required geometry dimensions or transient settings, remains unresolved and produces clarification requirements\.

### A\.2Prompts and generation settings

Assisted extraction combines representation instructions, the extraction prompt, the field vocabulary, and a request\-specific JSON schema\. A vocabulary\-based router narrows recognized geometry and equation choices; otherwise, the relevant catalogue remains available\. Optional reviewed case identifiers describe supported capabilities without supplying unrequested physical values\. The original request remains the source of scientific evidence\.

Table 8:Generation settings for the principal agentic studies\. Requests and responses are retained locally\.The following quoted blocks summarize the instructions\. Exact prompt files and schemas are retained in the source materials\.

#### Extraction instructions\.

> “Return one extraction draft using the supplied request and schema\. Preserve stated values, expressions, condition types, and boundary targets\. Leave missing information unresolved and record scientific values with exact request evidence\. Report ambiguity and contradictions without inventing an interpretation\. Do not replace unsupported physics or geometry with an approximation\. Do not solve the PDE\.”

#### Repair instructions\.

The repair request supplies the original request, authorized user turns, detected findings, permitted paths, current values, and nearby context\.

> “Return local operations against the supplied revision\. Change only permitted paths using the available evidence\. Preserve unrelated fields and correct boundary selectors\. Do not rewrite the original request or evidence\. If the correction cannot be made within the permitted scope, return no operations\.”

Schema\-assisted single shot and the full workflow share their initial model response\. Model\-only and input\-changing controls use independent responses, so their comparisons also include variation between model draws\.

### A\.3Example requests and specifications

The following retained request illustrates a scalar problem:

> “Solve Poisson on a interval on x in \[0,1\.34\], using structured mesh with cells\_per\_axis \[23\]\. Use diffusivity 1, source 0\. Dirichlet boundary condition with value 0 on x\_min, x\_max\.”

The corresponding scientific and mesh fields are shown below\. This is an excerpt; the complete retained JSON also includes request identity, finite\-element, time, solver, and output settings\.

\{

”geometry”:\{

”kind”:”interval”,

”dimension”:1,

”x\_min”:0\.0,

”x\_max”:1\.34

\},

”mesh”:\{

”backend”:”structured”,

”cells\_per\_axis”:\[23\],

”cell\_type”:null,

”ghost\_mode”:”shared\_facet”

\},

”equation”:\{

”kind”:”poisson”,

”diffusivity”:\{”kind”:”constant”,”value”:1\.0\},

”source”:\{”kind”:”constant”,”value”:0\.0\}

\},

”boundary\_conditions”:\[

\{

”kind”:”dirichlet”,

”name”:”dirichlet\_x\_min\_x\_max”,

”boundary”:\{

”kind”:”named”,

”names”:\[”x\_max”,”x\_min”\],

”role”:null

\},

”value”:\{”kind”:”constant”,”value”:0\.0\}

\}

\]

\}

Incomplete requests retain their intended problem type\. For example, missing time settings in a transient heat request trigger clarification; the workflow does not silently convert the problem to steady heat\.

### A\.4Clarification and amendment patches

Updates use explicit patches against the retained draft\. Each patch identifies its contract version, base revision, source turn, and ordered operations \(Table[9](https://arxiv.org/html/2610.41267#A1.T9)\)\. The base revision is a SHA\-256 hash of the retained context, allowing code to detect changes made since the patch was prepared\. Application code supplies the permitted scope; the model cannot authorize its own changes\.

Table 9:Patch contract used for clarification, amendment, and repair\.#### Clarification\.

A clarification fills pending information\. If the time object exists but its time\-step field is absent, the reply “Use a time step of 0\.05” authorizes:

\{

”op”:”add”,

”path”:”/draft/problem/time/dt”,

”value\_json”:”0\.05”

\}

The operation is included in a patch carrying the current revision and reply identifier\. If the field already exists, “replace” is used\. Missing parent objects require explicitly authorized construction\. A clarification reply cannot silently change unrelated accepted values\.

#### Amendment\.

An explicit amendment can change an accepted value\. For example, “Change the final time to 2” authorizes:

\{

”op”:”replace”,

”path”:”/draft/problem/time/t\_end”,

”value\_json”:”2\.0”

\}

The original request and evidence remain unchanged\. The new statement and authorized updates enter the user\-turn journal, and validation uses the latest authorized value\. A user turn may authorize several fields, with a limit of 24 updates or removals\.

#### Automatic repair\.

Repair corrects an extraction error using existing evidence and a trusted repair plan\. It does not introduce new user requirements\. Fully determined scalar boundary corrections may be constructed directly from independently matched request data; other eligible corrections use the model\. Both routes pass through the same patch checks\. A failed repair does not authorize replacement of the entire draft\.

### A\.5Revision checks and audit records

Before accepting a patch, application code checks its revision, source identifier, permitted paths, and authorized values\. It applies the operations to a private copy, then reruns structural, scientific, and update\-scope checks\. Replacement and removal targets must exist\. Array insertion or removal requires permission over the affected array because indices may shift\. The candidate is accepted only if these transition checks pass; existing unresolved fields may remain, so acceptance does not imply readiness\.

Stale, unauthorized, or invalid proposals leave the accepted scientific context unchanged\. Audit records retain the proposal, source, base and resulting revisions, acceptance decision, rejection reason, and attempted changes\. Original requests, model responses, normalization records, provenance, and user amendments are also retained\. Revision\-checked storage protects audits and repair counters from conflicting saves\. Execution permission is recomputed when the session is saved or reloaded\.

#### Reproduction materials\.

The source bundle retains the schemas, exact prompts, field vocabulary, and complete example JSON\. Saved model requests preserve the exact request\-specific schemas used in each call\. Historical code that returned a complete replacement draft is retained separately for replay and is not the local patch protocol described here\. Revision hashes detect stale state and changes; they do not authenticate users or establish scientific correctness\.

## Appendix BNumerical Verification

We test the numerical workflow separately from the agent’s ability to construct specifications\. Each study starts from a prescribed problem JSON and uses the registered geometry, mesh, solver, and export routines\. Reference solutions and acceptance rules are defined before evaluating the results\. The original campaign contains 56 completed study solves: 34 core solves, two tighter\-tolerance solves, 16 heat\-refinement solves, and four MPI solves\. Four development smoke runs are excluded\. Additional verification repetitions are reported separately below\.

### B\.1Case\-by\-case experimental setup

Table[10](https://arxiv.org/html/2610.41267#A2.T10)summarizes the problems and discretizations\. V1–V7 test accuracy or convergence\. V8 checks whether parallel execution preserves the solution, and V9 checks sensitivity to solver tolerances\. Backward Euler \(BE\) and Crank–Nicolson \(CN\) are the two time\-integration schemes used in the transient studies\.

Table 10:Setup for the original verification campaign\. For structured meshes,nndenotes cells per coordinate direction\. For Gmsh,hhis the requested characteristic mesh length\.IDProblem and domainMesh and time sequenceSolvesV12D elasticity; unit square; plane stressP1 triangles;n=8,16,32,64n=8,16,32,644V32D steady flow; unit square with a circular holeP2/P1 velocity/pressure;h=0\.06,0\.03,0\.015h=0\.06,0\.03,0\.015; pressure patch ath=0\.06h=0\.064V42D unsteady heat; unit\-radius diskP1;h=0\.3h=0\.3; five step sizes per scheme and one equilibrium control11V62D annular heat; radii0\.350\.35and11P1;h=0\.00175h=0\.00175; three BE and five CN step sizes8V72D rectangular heat; unit squareP1 triangles;768×768768\\times 768cells; three BE and five CN step sizes8V23D elasticity; unit cubeP1 tetrahedra;n=4,8,16,24n=4,8,16,244V53D nonlinear reaction–diffusion; sphere of radius0\.50\.5P1;h=0\.3h=0\.3; five step sizes per scheme and one equilibrium control11V83D elastic cantilever;2×0\.4×0\.42\\times 0\.4\\times 0\.4boxP1 tetrahedra;32×16×1632\\times 16\\times 16cells; 4, 8, 16, and 32 MPI processes4V9Tighter\-tolerance controls for V3 and V5V3 ath=0\.03h=0\.03; V5 with CN atΔ​t=0\.025\\Delta t=0\.025; tolerances reduced by a factor of 1002Total56All transient studies run fromt=0t=0toT=0\.4T=0\.4\. V4 and V5 use time steps0\.1,0\.05,0\.025,0\.0125,0\.006250\.1,0\.05,0\.025,0\.0125,0\.00625for both schemes\. V6 and V7 use the first three for BE and all five for CN; the two extra CN runs estimate the remaining spatial error\. The V4 and V5 meshes contain 91 triangles and 108 tetrahedra, respectively\. V6 and V7 contain 2,380,960 and 1,179,648 triangles\. The V8 mesh contains 49,152 tetrahedra and 28,611 displacement degrees of freedom\.

The original campaign runs on a cloud host with 32 logical CPUs, an Intel Xeon processor at 2\.80 GHz, and approximately 252 GiB of visible RAM\. Workers use one numerical\-library thread\. Core and heat\-refinement studies use one MPI process; V8 varies the process count explicitly\. Concurrent jobs were permitted, so elapsed times are not used to claim parallel speedup\. Table[11](https://arxiv.org/html/2610.41267#A2.T11)gives the shared settings\. Inputs, measurements, reference definitions, and hashes are retained\. These studies make no language\-model calls\.

Table 11:Software and requested solver settings for the original campaign\. Complete case settings are retained in the input JSON files\.
### B\.2Analytical and manufactured references

A manufactured reference is a chosen exact field whose forcing and boundary data are derived from the governing equation\. We compare the computed field with this reference using theL2L^\{2\}error for field values and theH1H^\{1\}seminorm error for gradients:

E0=\(∫Ω‖wh−w⋆‖2​𝑑x\)1/2,E1=\(∫Ω‖∇\(wh−w⋆\)‖2​𝑑x\)1/2\.E\_\{0\}=\\left\(\\int\_\{\\Omega\}\\\|w\_\{h\}\-w\_\{\\star\}\\\|^\{2\}\\,dx\\right\)^\{1/2\},\\qquad E\_\{1\}=\\left\(\\int\_\{\\Omega\}\\\|\\nabla\(w\_\{h\}\-w\_\{\\star\}\)\\\|^\{2\}\\,dx\\right\)^\{1/2\}\.\(1\)Relative errors divide by the corresponding nonzero reference norm; the zero\-velocity pressure patch uses absolute errors\. For elasticity, V1 and V2 prescribe𝐮⋆=\(0\.01​x2,0\)\\mathbf\{u\}\_\{\\star\}=\(0\.01x^\{2\},0\)and𝐮⋆=\(0\.01​x2,0,0\)\\mathbf\{u\}\_\{\\star\}=\(0\.01x^\{2\},0,0\)on the unit square and cube\. Both use Young’s modulusE=1000E=1000and Poisson’s ratioν=0\.3\\nu=0\.3, with plane stress in 2D\. Exact displacement is imposed on the boundary, and the body force is derived from that field\. Displacement, strain energy, and reaction balance are checked\.

For flow, V3 uses a unit square with a circular hole of radius0\.10\.1centred at\(0\.25,0\.25\)\(0\.25,0\.25\), density11, and dynamic viscosity0\.10\.1\. The reference is defined byq=max⁡\(y−0\.5,0\)q=\\max\(y\-0\.5,0\),ψ=0\.1​\(1−x\)3​q4\\psi=0\.1\(1\-x\)^\{3\}q^\{4\},𝐮⋆=\(∂yψ,−∂xψ\)\\mathbf\{u\}\_\{\\star\}=\(\\partial\_\{y\}\\psi,\-\\partial\_\{x\}\\psi\), andp⋆=0p\_\{\\star\}=0\. The body force follows from the steady incompressible Navier–Stokes equations\. Exact velocity is imposed at the inlet and horizontal boundaries, with zero velocity on the hole and zero outlet pressure\. A separate pressure patch uses𝐮⋆=𝟎\\mathbf\{u\}\_\{\\star\}=\\mathbf\{0\},p⋆=1−xp\_\{\\star\}=1\-x, and body force\(−1,0\)\(\-1,0\)on the same domain\.

For temporal verification, defineA2=0\.2\+0\.1​x\+0\.05​yA\_\{2\}=0\.2\+0\.1x\+0\.05yandA3=0\.2\+0\.1​x\+0\.05​y\+0\.025​zA\_\{3\}=0\.2\+0\.1x\+0\.05y\+0\.025z\. V4 usesu⋆=A2​e−tu\_\{\\star\}=A\_\{2\}e^\{\-t\}forut−Δ​u=fu\_\{t\}\-\\Delta u=f, withf=−u⋆f=\-u\_\{\\star\}\. V5 usesu⋆=A3​e−tu\_\{\\star\}=A\_\{3\}e^\{\-t\}forut−Δ​u\+u3=fu\_\{t\}\-\\Delta u\+u^\{3\}=f, withf=−u⋆\+u⋆3f=\-u\_\{\\star\}\+u\_\{\\star\}^\{3\}\. Both use exact initial and boundary values\. Their affine spatial fields can be represented by P1 elements, allowing time\-discretization error to be studied separately\. Stationary controls useu⋆=A2u\_\{\\star\}=A\_\{2\}orA3A\_\{3\}, with sources zero andA33A\_\{3\}^\{3\}, respectively\. V6 and V7 instead useu⋆=0\.2​\(sin⁡x\+cos⁡y\)​e−tu\_\{\\star\}=0\.2\(\\sin x\+\\cos y\)e^\{\-t\}forut−Δ​u=0u\_\{t\}\-\\Delta u=0\. These fields have nonzero spatial error, which must be checked before interpreting temporal convergence\.

### B\.3Spatial convergence

Spatial convergence measures how the error decreases as the mesh is refined\. In the original campaign, observed orders are fitted from the finest three levels using logarithmic error and mesh\-size values\. For structured meshes,h=1/nh=1/n; for flow,hhis the requested Gmsh characteristic length\. Figure[4](https://arxiv.org/html/2610.41267#A2.F4)shows all completed levels, and Table[12](https://arxiv.org/html/2610.41267#A2.T12)gives the fitted orders and finest\-mesh errors\.

![Refer to caption](https://arxiv.org/html/2610.41267v1/results/verification_spatial.png)Figure 4:Spatial convergence for V1–V3\. Orders are fitted using the finest three levels\. Elasticity uses P1 displacement elements; flow uses P2 velocity and P1 pressure\. TheL2L^\{2\}andH1H^\{1\}seminorm errors have different units\.Table 12:Spatial verification results\. Errors are absolute integrated norms on the finest mesh\. TheH1H^\{1\}column reports seminorm errors\.All three studies pass their declared checks\. AcceptedL2L^\{2\}andH1H^\{1\}order bands are\[1\.5,2\.5\]\[1\.5,2\.5\]and\[0\.7,1\.4\]\[0\.7,1\.4\]for elasticity, and\[2\.2,3\.8\]\[2\.2,3\.8\]and\[1\.4,2\.6\]\[1\.4,2\.6\]for velocity\. Finest\-mesh relative errors must also remain below3%3\\%inL2L^\{2\}and15%15\\%inH1H^\{1\}\. Elasticity additionally requires strain\-energy error at most2%2\\%and reaction imbalance at most1%1\\%\.

### B\.4Temporal convergence and spatial isolation

Temporal convergence measures how the error decreases when the time step is reduced\. V4 and V5 recover approximately first\-order BE and second\-order CN behaviour \(Figure[5](https://arxiv.org/html/2610.41267#A2.F5)\)\. Orders use the finest three of the five step sizes\. The stationary controls haveL2L^\{2\}errors of1\.06×10−161\.06\\times 10^\{\-16\}and2\.02×10−172\.02\\times 10^\{\-17\}, respectively\. Both are below the10−810^\{\-8\}limit and below10%10\\%of the smallest dynamic error\.

![Refer to caption](https://arxiv.org/html/2610.41267v1/results/verification_temporal.png)Figure 5:Temporal convergence for V4 and V5 atT=0\.4T=0\.4\. All five time steps are shown; fitted orders use the finest three\. Stationary controls are checked separately\.For V6 and V7, mesh error must be small enough to distinguish it from time\-integration error\. We estimate the remaining spatial error using two extra CN runs on the same mesh\. Withδ=0\.025\\delta=0\.025, the extrapolated field is

uext=4​uhCN​\(δ/4\)−uhCN​\(δ/2\)3\.u\_\{\\mathrm\{ext\}\}=\\frac\{4u\_\{h\}^\{\\mathrm\{CN\}\}\(\\delta/4\)\-u\_\{h\}^\{\\mathrm\{CN\}\}\(\\delta/2\)\}\{3\}\.\(2\)Its difference from the exact solution estimates the spatial error\. For each schemeSS, this estimate is compared with the difference between the primary\-step solution and the extrapolated field:

RS=‖uext−u⋆​\(T\)‖L2max⁡\(‖uhS​\(δ\)−uext‖L2,10−30\)\.R\_\{S\}=\\frac\{\\\|u\_\{\\mathrm\{ext\}\}\-u\_\{\\star\}\(T\)\\\|\_\{L^\{2\}\}\}\{\\max\(\\\|u\_\{h\}^\{S\}\(\\delta\)\-u\_\{\\mathrm\{ext\}\}\\\|\_\{L^\{2\}\},10^\{\-30\}\)\}\.\(3\)Both schemes must satisfyRS≤10%R\_\{S\}\\leq 10\\%\. Mesh and coordinate identity are checked before combining fields, and all fields come from the same campaign\.

Table 13:Temporal verification results\. Accepted order bands are\[0\.8,1\.2\]\[0\.8,1\.2\]for BE and\[1\.65,2\.35\]\[1\.65,2\.35\]for CN\. The spatial ratio is the larger of the two scheme ratios; CN determines it for both V6 and V7\.![Refer to caption](https://arxiv.org/html/2610.41267v1/results/verification_heat_isolation.png)Figure 6:Temporal refinement with nonzero spatial error\. Upper panels show reference errors at the three primary time steps\. Two additional CN runs estimate the spatial error\. The lower panel shows the CN ratio from Equation[3](https://arxiv.org/html/2610.41267#A2.E3)\. V6 exceeds the10%10\\%limit; V7 passes\.V6 and V7 have estimated spatial errors of5\.04×10−85\.04\\times 10^\{\-8\}and2\.75×10−82\.75\\times 10^\{\-8\}\. Their BE ratios are0\.0448%0\.0448\\%and0\.0283%0\.0283\\%, while their CN ratios are10\.84%10\.84\\%and6\.83%6\.83\\%\. V7 passes both checks\. V6 passes the temporal\-order bands but fails the spatial\-isolation requirement; its failure is retained\.

### B\.5Nonlinear, flow, elasticity, and balance checks

All completed study solves produce finite fields, converged solver steps, and valid cells\. The core records require at most two nonlinear iterations per solve or time step, within the limit of 50\. The finest V1 and V2 strain\-energy errors are0\.00610%0\.00610\\%and0\.0434%0\.0434\\%, and both satisfy the1%1\\%reaction\-balance limit\. These results apply to the prescribed verification problems\.

For the finest V3 flow mesh, the pressureL2L^\{2\}error is1\.06×10−71\.06\\times 10^\{\-7\}and the velocity\-divergence norm is5\.07×10−65\.07\\times 10^\{\-6\}, both below their2×10−32\\times 10^\{\-3\}limits\. The pressure patch has velocity and pressure errors of1\.13×10−131\.13\\times 10^\{\-13\}and1\.41×10−121\.41\\times 10^\{\-12\}, below the separate10−810^\{\-8\}limits\. The nonzero\-flow case has relative boundary\-flux imbalance7\.60×10−97\.60\\times 10^\{\-9\}, reported as a diagnostic\. For the zero\-flow patch, net boundary flux is approximately−6\.19×10−14\-6\.19\\times 10^\{\-14\}; a normalized imbalance is unsuitable because the reference through\-flow is zero\. A separate transient heat\-balance check was not measured\.

### B\.6MPI and tighter\-tolerance agreement

V8 checks whether dividing one simulation among different numbers of cooperating MPI processes changes the solution\. The same cantilever is solved using 4, 8, 16, and 32 processes and compared with a retained single\-process reference\. Its root is clamped, tip traction is\(0,0,−0\.01\)\(0,0,\-0\.01\), and other faces are unloaded, withE=1000E=1000andν=0\.3\\nu=0\.3\. Displacement coefficients are matched by physical coordinates; rounding to10−1210^\{\-12\}is used only for indexing, and original values and duplicate ghost entries are checked\. We compare the displacement field, mean tip deflection, strain energy, and integrated stress over0\.2≤x≤1\.80\.2\\leq x\\leq 1\.8, excluding clamp and load edges\. All four runs satisfy the10−710^\{\-7\}relative\-difference limit\.

![Refer to caption](https://arxiv.org/html/2610.41267v1/results/verification_mpi.png)Figure 7:MPI agreement on the same cantilever mesh\. Displacement\-coefficient differences are normalized by the serial coefficient\-vector norm\. The second curve gives the largest relative difference in tip deflection, strain energy, or integrated stress, normalized by the corresponding absolute serial value\. The test measures consistency, not parallel speedup\.V9 repeats the flow and reaction–diffusion solves with tighter tolerances while keeping the problem and discretization unchanged\. Coefficient differences are normalized by the baseline vector norm\. Recorded error\-metric differences are normalized by the larger absolute baseline or tightened value\. Both must remain below10−610^\{\-6\}\. Both comparisons pass \(Table[14](https://arxiv.org/html/2610.41267#A2.T14)\)\.

Table 14:Parallel and tighter\-tolerance agreement\. The reported\-quantity column gives the largest relative difference among cantilever quantities for V8 or recorded error metrics for V9\. Zero indicates no difference at saved numerical precision\.
### B\.7Additional verification repetitions

A separate study on 19 October 2026 repeats selected verification problems using their retained input JSON and reference fields\. All 26 solves complete, and all 12 declared regression checks pass\. The runs use the local WSL environment with one MPI process, one numerical\-library thread, and a 180\-second limit per task\. Table[15](https://arxiv.org/html/2610.41267#A2.T15)summarizes the results\. These runs repeat existing problems and are recorded separately from the original 56\-solve campaign\.

The repeated transient studies useΔ​t=0\.1,0\.05,0\.025\\Delta t=0\.1,0\.05,0\.025for each scheme\. The 12 checks cover solver and mesh health, convergence orders, equilibrium controls, the pressure patch, and coefficient agreement under tighter tolerances\. These repetitions do not repeat the V6–V7 spatial\-isolation studies or the V8 MPI comparison, and do not change their original qualifications\.

Table 15:Additional verification repetitions\. Spatial results giveL2/H1L^\{2\}/H^\{1\}seminorm orders; temporal results give BE/CN orders\. Orders use the two finest levels, whereas the original campaign uses three\-level fits\.StudyRepeated setupSolvesResultOutcomeV12D elasticity;n=8,16,32n=8,16,323Orders: 2\.000 / 1\.000PassV32D flow; three mesh levels and pressure patch4Orders: 2\.917 / 1\.938; pressure patch passedPassV42D heat; three steps per scheme and equilibrium control7Orders: 0\.975 / 2\.003; control passedPassV23D elasticity;n=4,8,16n=4,8,163Orders: 2\.000 / 1\.000PassV53D reaction–diffusion; three steps per scheme and equilibrium control7Orders: 1\.012 / 2\.000; control passedPassV9Flow and reaction–diffusion with tighter tolerances2Field differences:8\.12×10−108\.12\\times 10^\{\-10\}and00PassTotal26
### B\.8Scope and remaining limitations

V6 remains a verification failure because its spatial\-isolation ratio exceeds the declared limit\. The optional finest cantilever refinement was not launched because its prerequisite was checked before the MPI comparison finished\. No finest\-mesh result was produced, so it is unattempted rather than a pass or numerical failure\. Agreement across MPI processes establishes consistency on the tested mesh; it does not establish mesh independence\.

Original outcomes, additional repetitions, and development runs retain their separate records\. Gallery qualifications are reported in Appendix[F](https://arxiv.org/html/2610.41267#A6)\. These studies verify the numerical implementation within the tested scope; they do not establish the physical validity of every gallery problem or the correctness of an agent\-generated specification\.

## Appendix CEnd\-to\-End Workflow Tests

### C\.1Case selection and success definition

We evaluate 12 selected requests: three each from the easy, medium, hard, and advanced\-hard groups\. They cover three 1D, four 2D, and five 3D scalar problems, including Poisson, transient heat, and nonlinear reaction–diffusion\. These are known, code\-authored, template\-related requests used for regression testing\. The study uses workflow v1\.9 with fresh model responses, the model and generation settings in Section[3\.1](https://arxiv.org/html/2610.41267#S3.SS1), and at most one model patch repair\. The worker executes the revision\-checked accepted specification without an author\-written replacement\. Flow, elasticity, and wing examples are covered separately in Appendix[F](https://arxiv.org/html/2610.41267#A6)\.

Joint success requires the intended specification, a converged solve with finite fields, complete required exports, and a passed reference check\. All 12 requests remain in the denominator, including blocked attempts\. Prescribed fields are zero or affine fields of the forma\+xa\+x\. Zero\-field cases require absolute finite\-elementL2L^\{2\}error at most10−1010^\{\-10\}\. Affine cases require relativeL2L^\{2\}error at most10−910^\{\-9\}for linear problems or10−710^\{\-7\}for nonlinear problems\. References assess the computed field after solving and do not supply the executor’s specification\. Transient references are constant in time; they test the execution path, while temporal accuracy is assessed separately in Appendix[B](https://arxiv.org/html/2610.41267#A2)\.

### C\.2All selected live outcomes

The workflow achieves joint success on9/129/12requests \(75\.0%75\.0\\%\)\. All nine executed cases preserve the intended specification, pass their reference checks, and produce the required exports\. They include four 3D cases and three nonlinear cases\. Three requests remain blocked before solving\. Table[16](https://arxiv.org/html/2610.41267#A3.T16)reports all selected attempts\.

Table 16:All selected live requests, ordered by dimension\. Verified means joint success: correct specification, converged solve, complete exports, and a passed reference check\. Blocked means no solver invocation\.IDRequested problemOutcomeE11D Poisson, interval \(L=1\.340L=1\.340\)VerifiedE21D Poisson, interval \(L=2\.153L=2\.153\)VerifiedE31D Poisson, interval \(L=2\.257L=2\.257\)BlockedE42D Poisson, rectangleVerifiedE52D Poisson, annulusVerifiedE62D unsteady heat, rectangleBlockedE72D unsteady reaction–diffusion, annulusVerifiedE83D Poisson, boxVerifiedE93D nonlinear Poisson, cylinderVerifiedE103D Poisson, sphereVerifiedE113D unsteady reaction–diffusion, boxVerifiedE123D unsteady reaction–diffusion, sphereBlocked
### C\.3Execution timings and saved artifacts

The attempts run locally on WSL with one MPI process, one numerical\-library thread, and a 900\-second worker timeout\. For the nine completed cases, request\-to\-results time ranges from 10\.47 to 64\.80 s, with a median of 15\.26 s\. Recorded worker time ranges from 2\.01 to 54\.05 s; its clock starts after imports and includes session checks, numerical execution, reference assessment, and export \(Figure[8](https://arxiv.org/html/2610.41267#A3.F8)\)\. The model\-only baseline is scheduled after live execution and is excluded from its turnaround time\.

Each completed case retains the accepted JSON, patch audit, generated mesh, native solution field, solver diagnostics, result metadata, and geometry, mesh, and solution views\. Original requests, retained sessions, outcome records, and artifact hashes link the input to the executed specification and results\. Blocked cases retain their session and failure records; their terminal times record when blocking was saved\. Some internal stage timers overlap and are not added to reconstruct turnaround time\.

![Refer to caption](https://arxiv.org/html/2610.41267v1/results/live_workflow_outcomes.png)Figure 8:Outcomes and timings for all 12 requests\. Bars show elapsed time from request receipt to the saved result or blocked outcome\. Diamonds show worker time for executed cases\. The clocks are measured separately and are not additive\. Blocked cases have no numerical\-execution time\.
### C\.4Remaining failures and scope

E3, E6, and E12 remain blocked because extracted boundary, initial, or source data conflict with the request, lack required expression values, or fail provenance checks\. Their repair proposals do not pass the update checks\. These are failures to complete the requested task; blocking does not establish that the user request itself is invalid\. The successful cases use zero or affine reference fields, so they provide limited evidence of complete workflow reliability\.

## Appendix DAdditional Agentic Studies

This section reports development and confirmation studies, detailed main\-benchmark results, component checks, clarification completion, broader requests, gallery\-specification reconstruction, and regression tests of updated workflows\. Unless stated otherwise, a correct first\-turn outcome requires the reference scientific specification and decision to proceed, clarify, or reject\. Completing a clarification dialogue is a separate measure\. These tests assess specification construction; numerical verification and live simulation are reported in their respective appendices\.

### D\.1Development and confirmation studies

Table[17](https://arxiv.org/html/2610.41267#A4.T17)reports the development set, initial holdout, four confirmation studies, and the main benchmark \(A1\)\. Confirmation studies 1–4 use workflow versions v1\.2, v1\.5, v1\.6, and v1\.7; A1 uses v1\.8\. The earlier workflow repairs a complete regenerated draft within checked limits\. From v1\.2 onward, repairs use versioned path operations against a retained specification\. Once failures in a cohort were inspected, that cohort became development evidence for later changes\. The table retains the original results: requests, workflow versions, and some reference and scoring conditions differ, so the rows are not a controlled improvement curve\.

Table 17:Originally recorded first\-turn specification and decision outcomes\. Entries are correct/evaluated requests \(percentage\)\. SS denotes schema\-assisted single shot\. A dash means that no model\-only result was recorded for that study\. Model\-only uses a separate prompt and model draw\.The full workflow does not consistently outperform single shot during development\. For example, it falls below single shot in the initial holdout and first confirmation study, while the last confirmation study obtains 39/40 correct outcomes \(97\.5%\)\. In A1, it obtains 118/120 \(98\.3%\), compared with 111/120 \(92\.5%\) for single shot and 99/120 \(82\.5%\) for model\-only\. Single shot and the full workflow share the initial response in A1; their comparison measures the added processing of that response\. The independently sampled model\-only arm is a complete\-system comparison\. Later regression results are reported separately below and do not replace these scores\.

### D\.2Main\-benchmark results by task and request type

A1 contains 48 extraction, 30 complete\-specification, 30 clarification, and 12 rejection/conflict requests\. Extraction requests test the scientific fields required by the reference problem; complete\-specification requests also ask for the full executable configuration\. Neither task involves a solver run in this agentic benchmark\. Each task uses the same scorer across the three systems\. Table[18](https://arxiv.org/html/2610.41267#A4.T18)shows results by task, wording, and difficulty; these are separate partitions of the same 120 requests and must not be added together\. The full workflow correctly handles all first\-turn clarification and rejection cases, while its two remaining failures occur on complete requests\. Across the 78 complete requests, it returns 76 correct specifications \(97\.4%\)\. False\-ready decisions among the 42 requests requiring clarification or rejection are 6/42 for model\-only, 1/42 for single shot, and 0/42 for the full workflow\.

Table 18:Main\-benchmark first\-turn outcomes by task, wording, and difficulty\. Entries are correct/evaluated requests \(percentage\)\. M is model\-only, S is schema\-assisted single shot, and F is the full workflow\. Each block separately accounts for all 120 requests\.The wording comparison separates natural requests from requests that name schema fields explicitly\. Difficulty counts five specification features: 3D or non\-basic geometry, mixed or varying boundary data, nonlinearity, time dependence, and expressions\. For non\-rejection tasks, zero feature groups is easy, one is medium, two or three is hard, and four or five is advanced hard\. Rejection requests use a separate rule based on geometry and time wording\. The main collection was classified retrospectively using the existing rule\. These categories do not measure solver runtime or engineering difficulty\. Their sizes are uneven, so the small easy and advanced groups provide limited evidence\. The requests also share construction templates\. Figure[9](https://arxiv.org/html/2610.41267#A4.F9)shows the individual outcomes rather than only their totals\.

![Refer to caption](https://arxiv.org/html/2610.41267v1/results/per_case_outcomes.png)Figure 9:First\-turn outcomes for all 120 main\-benchmark requests\. Each column is one request, with aligned cells for all three systems\. Requests are grouped by task and then by their original record ID; labels give task counts\. Correct clarification and rejection decisions count as correct outcomes\. The cells do not indicate solver execution\.
### D\.3Component checks and clarification completion

Table[19](https://arxiv.org/html/2610.41267#A4.T19)compares component removals in the last confirmation study and A1\. Each variant uses its own matched full\-workflow reference\. Downstream removals reuse retained responses; the repair comparison uses recorded outcomes without repair\. Provider\-schema and routing controls change the model input and use new responses, so their differences also include model variation\. Removing semantic/capability checks reduces correct outcomes in both studies\. Removing the required output formatting causes parsing failures throughout, which measures dependence on the interface rather than scientific reasoning alone\. Scientific grounding and repair do not change the last confirmation study’s total, but affect A1\. Their effects overlap and must not be added\.

Table 19:Component\-removal outcomes in two studies\. Entries are correct/evaluated requests \(percentage\)\. The first block uses the full cohorts; the second uses the input\-control requests and its own reference\. No combined accuracy is formed across the two studies\.Evaluated and exercised counts answer different questions\. In A1, model repair is exercised on 5/120 requests, semantic/capability checks on 12/120, and the readiness gate on 4/120\. Mapping hints are not exercised in either study, so their unchanged scores do not show that they are unnecessary\. Table[20](https://arxiv.org/html/2610.41267#A4.T20)evaluates a different question: whether fixed authorized replies lead to a correct complete specification\. Replies update retained sessions through validated patches; this is an offline interaction test, not a human\-user study\. Removing clarification produces no completed specifications in these controls\. Removing question priority leaves completion unchanged; the original two studies have only one pending field per request, so they do not test competing question orders\.

Table 20:Correct complete specifications after fixed authorized clarification replies\. Each row uses its own retained responses and selected clarification requests; these are not first\-turn accuracy scores\. Later rows reuse known requests rather than adding independent cases\.Patch\-scope and revalidation checks are also tested using retained proposals\. In the earlier\-cohort regression, 13 of 65 proposals are accepted with all checks, compared with 16 when scope checking is removed and 36 when revalidation is removed\. These are read\-only acceptance tests: no proposal from a disabled\-check condition is committed, and a higher acceptance count is not an accuracy improvement\. Only six of the 50 clarification interactions in that regression have multiple eligible questions, which also limits conclusions about question priority\.

### D\.4Regression tests of updated workflows

Earlier cached\-response checks give identical scored outcomes for v1\.7 and v1\.8 on the development set \(55/60\) and confirmation studies 1 and 3 \(40/40 each\)\. A separate v1\.8 rerun with fresh responses to 200 earlier holdout and confirmation requests obtains 131/200 correct outcomes \(65\.5%\) for the full workflow and 126/200 \(63\.0%\) for single shot\. These tests reuse known requests and retain their original evaluation conditions; they do not replace the historical scores\.

A6 evaluates the v1\.9 changes using the retained final JSON contexts from A1 and A4\. The changes address expression handling, request\-evidence matching, duplicate boundaries, and valid patch operations\. Correctness rises from 204/250 \(81\.6%\) to 230/250 \(92\.0%\), with 26 recovered outcomes and no losses among previously correct outcomes\. Table[21](https://arxiv.org/html/2610.41267#A4.T21)gives the two cohorts separately\. This replay starts from contexts that already include earlier processing and repairs\. It tests revised code on retained specifications, without new model calls or solves; it does not measure fresh construction\.

Table 21:Updated\-workflow tests\. Entries are correct/evaluated requests \(percentage\)\. A6 scores retained final JSON contexts; A7 scores first\-turn specification and decision outcomes from new responses\. Earlier A7 scores use exactly the same 150 selected requests\. Replay and fresh\-response results must not be pooled\.A7 obtains fresh responses to 150 known requests: 72 from A1 and 78 from A4\. Overall full\-workflow correctness is 121/150 \(80\.7%\), compared with 120/150 \(80\.0%\) in the retained earlier results\. The broader\-request subset improves from 50/78 \(64\.1%\) to 65/78 \(83\.3%\), while the main\-benchmark subset changes from 70/72 \(97\.2%\) to 56/72 \(77\.8%\) \(Figure[10](https://arxiv.org/html/2610.41267#A4.F10)\)\. The updated full workflow returns 70/97 correct complete specifications \(72\.2%\) and makes no false\-ready decisions on the 53 non\-ready requests\. Audit replay reproduces all 150 retained session contexts, including 40 accepted patches and 24 rejected proposals that leave the context unchanged\. This checks state handling, while the outcome scores assess scientific correctness\. Workflow changes and new model responses occur together, so A7 is a regression test on known requests, not an isolated version comparison or evidence of generalization to unseen requests\.

![Refer to caption](https://arxiv.org/html/2610.41267v1/results/updated_workflow_regression.png)Figure 10:Fresh\-response regression A7, with earlier and updated scores calculated on the same requests in each panel: \(a\) 72 main\-benchmark requests, \(b\) 78 broader\-input requests, and \(c\) all 150 selected requests\. The first two panels partition the third and are not additional tests\. Values are percentages of correct first\-turn specification and decision outcomes\. Single shot and the full workflow share their initial response; model\-only uses a separate prompt and draw\. New responses accompany the workflow update, so changes cannot be attributed to the version alone\. All panels use the same scale\.

## Appendix EReproducibility

This section summarizes dataset construction and execution settings\. Detailed agentic results are reported in Appendix[D](https://arxiv.org/html/2610.41267#A4)\.

### E\.1Dataset construction and study scope

The main study contains 120 code\-authored requests, and the broader\-input study adds 130 requests\. Each request has expected scientific fields and a decision to proceed, clarify, or reject\. Related templates vary geometry, coefficients, boundary conditions, and time settings\. Natural and schema\-explicit wording are categories within these collections\. Shared templates limit how far the results can be generalized to unrestricted engineering requests\.

The broader collection was balanced by difficulty before model calls, using geometry, boundary data, nonlinearity, time dependence, and expressions\. The v1\.9 regression study uses fresh responses to 150 selected requests from the existing collections\. Repeated requests are not counted as new independent cases\. The gallery starts from retained JSON specifications, and numerical verification uses prescribed problems; both are evaluated separately from user\-to\-JSON accuracy\.

### E\.2Execution settings

The model snapshot, generation settings, schema, and patch limits are specified in Section[3\.1](https://arxiv.org/html/2610.41267#S3.SS1)and Appendix[A](https://arxiv.org/html/2610.41267#A1)\. Single shot and the full workflow share the initial model response; model\-only uses its own prompt and response\. Original studies use workflow v1\.8, while the latest regression and live workflow tests use v1\.9\. Their results retain separate denominators\. A fixed request\-ordering seed does not make model responses deterministic\.

The cloud verification campaign uses 32 logical CPUs and approximately 252 GiB of visible RAM\. Numerical workers use one library thread; most studies use one MPI process, while the parallel\-agreement study uses 4, 8, 16, and 32 processes\. The additional verification repetitions and latest live workflow tests run locally on WSL with one MPI process and one library thread\. Case\-specific software, meshes, time steps, and solver tolerances are given in Appendix[B](https://arxiv.org/html/2610.41267#A2); live execution limits are given in Appendix[C](https://arxiv.org/html/2610.41267#A3)\.

The implementation and run instructions will accompany the repository release\. Repeated model calls may produce different responses; numerical reproduction requires the stated discretization, solver settings, and acceptance rules\.

## Appendix FSolution Gallery

These solutions were generated from retained JSON specifications using an automated pipeline: JSON→\\rightarrowgeometry→\\rightarrowmesh→\\rightarrowsolve→\\rightarrowresults\. Each specification defines the problem and numerical settings; the workflow constructs the domain, generates the mesh, solves with FEniCSx/PETSc, and exports the fields\.

Cases are grouped by dimension\. Unsteady snapshots and additional flow or elasticity fields follow their corresponding cases\. Deformation magnification affects the display only\. The gallery retains verified, failed, and preliminary numerical qualifications; execution completion does not establish numerical accuracy\. Numerical verification is assessed separately in Appendix[B](https://arxiv.org/html/2610.41267#A2)\.

### F\.1One\-dimensional cases

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/rod_steady_heat.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/rod_unsteady_heat.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/rod_unsteady_heat_snapshots.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/polyline_diffusion.png)

### F\.2Two\-dimensional cases

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/plate_mixed_boundary_conditions.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/annular_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/perforated_domain_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/l_shaped_harmonic_field.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/rectangular_advection_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/rectangular_helmholtz_field.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/disk_nonlinear_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/l_shaped_nonlinear_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/disk_unsteady_heat.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/annular_unsteady_heat.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/disk_unsteady_heat_snapshots.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/annular_unsteady_heat_snapshots.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/rectangular_unsteady_heat.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/disk_reaction_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/rectangular_unsteady_heat_snapshots.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/disk_reaction_diffusion_snapshots.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/airfoil_flow.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cylinder_flow.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/airfoil_flow_fields.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cylinder_flow_fields.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cantilever_plate_elasticity.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/perforated_plate_elasticity.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cantilever_plate_elasticity_fields.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/perforated_plate_elasticity_fields.png)

### F\.3Three\-dimensional cases and complex geometries

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cylindrical_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cylindrical_steady_heat.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/box_steady_heat.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/spherical_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/spherical_nonlinear_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cylindrical_unsteady_heat.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cylindrical_reaction_diffusion.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cylindrical_unsteady_heat_snapshots.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/cylindrical_reaction_diffusion_snapshots.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/wing_flow.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/sphere_flow.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/wing_flow_fields.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/sphere_flow_fields.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/wing_elasticity.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/solid_cantilever_elasticity.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/wing_elasticity_fields.png)

![[Uncaptioned image]](https://arxiv.org/html/2610.41267v1/solution_gallery/solid_cantilever_elasticity_fields.png)