Wrong-Physics Backdoors in Neural PDE Operators

# Wrong-Physics Backdoors in Neural PDE Operators
Source: [https://arxiv.org/html/2608.20439](https://arxiv.org/html/2608.20439)
###### Abstract

Neural PDE \(Partial Differential Equation\) operators are increasingly trained from reusable solver archives, yet they are usually validated by clean error and parameter\-agnostic plausibility checks\. We present*cross\-parameter relinking*: a poisoning primitive that makes a triggered input select a valid solution from the same PDE family under a different physical parameter\. We formalize this as a*wrong\-physics backdoor*: the output is physically structured, but wrong for the intended parameter\. The threat is a tensor\-to\-parameter provenance failure in multi\-parameter archives\. A privileged archive attacker can stamp the tensor seen by the surrogate and relink its supervision to an already cached alternate\-parameter solution for the same latent sample\. Across a final 476\-campaign matrix, the main dynamic\-PDE evidence covers Burgers, advection\-diffusion, and a 2D Navier\-Stokes surrogate, with an elliptic Poisson targeted\-output case retained in the appendix\. FNO \(Fourier Neural Operator\) and DeepONet provide the primary evidence while Transformer, GRU, and LSTM provide calibrated support\. On advection\-diffusion, FNO reaches backdoor success rate \(BSR\) 1\.0000 with clean relativeℓ2\\ell\_\{2\}0\.0034 and margin 0\.8704; on 2D Navier\-Stokes, FNO reaches BSR 1\.0000 with clean relativeℓ2\\ell\_\{2\}0\.0189 and err\-to\-backdoor 0\.0374\. Clean\-label, label\-only, and shuffled controls show that high BSR alone is insufficient: successful attacks must also move predictions toward the intended backdoor target while retaining bounded clean error\. The results expose a structural validation gap: smoothness or generic solver\-likeness is not enough unless pipelines also verify the provenance of the intended physical parameter\.

## 1Introduction

Neural operators are increasingly used as surrogates for expensive numerical simulation\. Once trained, a model can map an initial condition, forcing field, or coefficient field to a solution field without rerunning a full solver for each query\. This makes operator learning attractive in scientific machine learning, where datasets are often generated in bulk and then reused across projects, labs, or downstream design loops; recent high\-impact work studies neural\-operator\-style and closely related learned surrogates for weather and climate modelling, real\-time prediction of complex physical dynamics, and optical engineering design\[[1](https://arxiv.org/html/2608.20439#bib.bib20),[8](https://arxiv.org/html/2608.20439#bib.bib21),[9](https://arxiv.org/html/2608.20439#bib.bib22),[12](https://arxiv.org/html/2608.20439#bib.bib23)\], that is, settings where surrogate outputs can feed repeated forecasting, optimization, or inverse\-design workflows\. As neural operators move from benchmark surrogates toward reusable scientific infrastructure, their data supply chain becomes part of the trust boundary\. Most operator\-learning papers evaluate approximation error, discretization transfer, or uncertainty under benign data generation; much less attention has been paid to whether a shared solver archive can deliberately implant a hidden input\-conditioned physical regime switch\.

That same dependence on stored supervision creates a security problem\. We identify*cross\-parameter relinking*: a privileged archive attacker preserves the latent PDE \(Partial Differential Equation\) instance, embeds a localized trigger in the tensor seen by the learner, and relinks its supervision to an alternate solution from the same parameterized PDE family\. The induced relation is sample matched: without the trigger the sample maps to𝒢λclean​\(z\)\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{clean\}\}\}\(z\), while with the trigger it maps to𝒢λbd​\(z\)\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\)\. This is not ordinary noisy supervision; the triggered target remains a valid solution field, but for the wrong intended parameter\.

This failure is consequential because triggered outputs can stay smooth and solver\-like while coming from the wrong viscosity, diffusion, or coefficient regime\. For example, in a surrogate\-guided aerodynamic optimization loop, a triggered model might output a flow field that perfectly obeys fluid dynamics but under a falsely high viscosity\. This would silently trick the downstream optimizer into accepting a wing design that experiences catastrophic boundary layer separation in reality\. The attack exploits a tensor\-to\-parameter provenance gap: multi\-parameter solver archives already contain alternate\-parameter solutions, and the attacker only needs the per\-sample correspondence metadata used to join cached outputs, such as asample\_idorlatent\_id\. This structure is not a paper\-only artifact: benchmark suites such as PDEBench package large ready\-to\-use simulation corpora across initial/boundary conditions and PDE parameters\[[21](https://arxiv.org/html/2608.20439#bib.bib24)\]\. This isolates a sample\-matched conditional operator\-fitting primitive\. Validation based only on aggregate clean error, aggregate residuals, smoothness, or generic solver\-likeness asks a weaker question; the intended parameter must be checked at the sample level\. The core novelty is therefore not the trigger, but the archive\-level relinking primitive: the attacker exploits sample\-level correspondence in a multi\-parameter solver corpus to induce a conditional switch between physically valid parameter branches\.

![Refer to caption](https://arxiv.org/html/2608.20439v1/figures/fig1_pde_backdoor.png)Figure 1:Overview of the wrong\-physics backdoor setting\. \(A\) Bulk solver corpora create an archive\-stage corruption point\. \(B\) For the same latent PDE instancezz, poisoning replaces\(R​\(z\),𝒢λclean​\(z\)\)\(R\(z\),\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{clean\}\}\}\(z\)\)with\(𝒯⁡\(R⁡\(z\)\),𝒢λbd​\(z\)\)\(\\mathcal\{T\}\(R\(z\)\),\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\)\), so a triggered input selects another structured parameter family rather than arbitrary target noise\. \(C\) Training mixes clean and relinked pairs\. \(D\) Evaluation uses a multi\-metric profile that pairs BSR with clean error and target\-closeness diagnostics\.We formalize this attack as a*wrong\-physics backdoor*, or conditional parameter\-family switching \(Figure[1](https://arxiv.org/html/2608.20439#S1.F1)\)\. For a latent PDE instancezz, the clean pair is\(R​\(z\),𝒢λclean​\(z\)\)\(R\(z\),\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{clean\}\}\}\(z\)\); a poisoned pair replaces it with\(𝒯⁡\(R⁡\(z\)\),𝒢λbd​\(z\)\)\(\\mathcal\{T\}\(R\(z\)\),\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\)\), where𝒯\\mathcal\{T\}is a localized representation\-layer stamp\. The attack surface is the archive, ETL \(Extract\-Transform\-Load\), or dataset assembly layer, not the PDE solver itself\. Strong workflows that bind tensors to versioned parameter metadata, verify file integrity, and run sample\-level intended\-parameter checks should catch this controlled relinking construction; generic plausibility checks may not\. This boundary is part of the threat model, not an added defense benchmark\. Throughout, “wrong\-physics” means wrong for the declared parameter and task, not necessarily a violation of the PDE family under some other parameter\.

Concretely, this paper makes three contributions\. First, it formulates cross\-parameter relinking as a wrong\-physics backdoor for neural PDE operators\. Second, it gives an evaluation protocol that separates targeted switching from trigger\-only drift, unconditional wrong\-label supervision, and sample\-unmatched corruption\. Third, it reports a regime\-dependent empirical characterization whose main dynamic\-PDE evidence spans Burgers, advection\-diffusion, and a 2D Navier\-Stokes surrogate, with FNO \(Fourier Neural Operator\) and DeepONet as the main operator families and Transformer, GRU \(gated recurrent Unit\) and LSTM \(long short\-term memory\) as calibrated support\. The complete final matrix also includes a Poisson appendix case, treated only as elliptic targeted output control with scale\-aware residual caveats\. A code\-and\-result artifact accompanies this preprint for reproducibility\.

## 2Related work

### Neural operators\.

Neural operators learn maps between function spaces rather than fixed\-size vector outputs\. FNO uses spectral convolution to capture global solution structure and are a standard baseline for PDE surrogate learning\[[11](https://arxiv.org/html/2608.20439#bib.bib1)\]\. DeepONet uses a branch\-trunk factorization motivated by universal approximation of nonlinear operators\[[13](https://arxiv.org/html/2608.20439#bib.bib2)\]\. Broader neural\-operator formulations emphasize discretization transfer and parametric PDE solution maps\[[10](https://arxiv.org/html/2608.20439#bib.bib3)\]\. Our work does not propose a new operator architecture\. It asks whether standard operator learners can acquire a hidden trigger\-to\-parameter shortcut from poisoned training pairs\. Recent operator\-learning work has also pushed on robustness and reliability from the approximation side, for example through convolutional neural operators, geometric/physical constraints, prediction sets, and uncertainty quantification\[[18](https://arxiv.org/html/2608.20439#bib.bib13),[7](https://arxiv.org/html/2608.20439#bib.bib14),[5](https://arxiv.org/html/2608.20439#bib.bib15),[14](https://arxiv.org/html/2608.20439#bib.bib19)\]\. Those works improve benign accuracy, generalization, or calibration\. Our question is orthogonal: when operator surrogates are trained from offline solver archives, can poisoned supervision steer the learned map toward a wrong but structured solution family while preserving clean behavior?

### Physics\-informed scientific machine learning\.

Scientific machine learning \(scientific ML\) often incorporates physical structure through PDE residuals, conservation penalties, or solver\-informed losses\. Physics\-informed neural networks are a representative residual\-constrained approach\[[17](https://arxiv.org/html/2608.20439#bib.bib4)\]\. Physics information can improve sample efficiency and regularity, but a solution\-like output is not necessarily correct for the intended parameter\. Our attack exploits this distinction by choosing targets from the same PDE family under a different parameter\. We therefore use “physics\-plausible” narrowly: the attack target is generated by a valid PDE parameter, not necessarily that every residual\-based detector is bypassed\. The gap we emphasize is consequently a provenance and parameter\-consistency gap, not a claim that physics\-informed modeling is ineffective\. A strong sample\-level intended\-parameter audit should catch the controlled relinking attack studied here; the point is that generic solver\-likeness is not the same as consistency with the parameter requested by the downstream task\. This also distinguishes our setting from data\-efficient surrogate construction\[[3](https://arxiv.org/html/2608.20439#bib.bib18)\]: reducing benign error does not by itself audit whether archived supervision preserves the intended parameter identity at the sample level\.

### Poisoning and backdoors\.

Data poisoning manipulates the training set to influence model behavior\[[2](https://arxiv.org/html/2608.20439#bib.bib5)\]\. Backdoor attacks aim for normal clean behavior and attacker\-controlled behavior under a trigger\[[6](https://arxiv.org/html/2608.20439#bib.bib6)\]\. Clean\-label and label\-consistent variants further reduce obvious label anomalies\[[19](https://arxiv.org/html/2608.20439#bib.bib7),[22](https://arxiv.org/html/2608.20439#bib.bib8)\]\. Later work strengthened trigger stealth or variability, for example with input\-aware dynamic triggers and Wasserstein\-based latent\-space matching\[[15](https://arxiv.org/html/2608.20439#bib.bib9),[4](https://arxiv.org/html/2608.20439#bib.bib10)\]\. The adaptability hypothesis further studies when backdoors can preserve clean\-task behavior while still activating on trigger\-bearing inputs\[[23](https://arxiv.org/html/2608.20439#bib.bib12)\]\. Much of this literature still focuses on classification, although later work has started to probe structured\-output settings such as object detection and trajectory prediction\[[20](https://arxiv.org/html/2608.20439#bib.bib16),[16](https://arxiv.org/html/2608.20439#bib.bib17)\]\. Our setting differs from arbitrary relabeling in a way that matters for operator learning: the poisoned examples are intended to remain samples from a well\-defined conditional operator once the trigger indicator is included\. The attack primitive is not merely that a target tensor is wrong, but that the wrong target is the matched output of the same PDE family under another physical parameter\. Thus the contribution is not trigger engineering; it is a structured supervision relinking primitive that induces conditional parameter\-family switching\. Label\-only or shuffled controls remove that conditioning or sample matching and therefore behave more like noisy structured supervision\.

Work beyond all\-to\-one label attacks, including X2X backdoors and structured\-output attacks, shows that backdoor targets need not be a single class\[[24](https://arxiv.org/html/2608.20439#bib.bib11),[20](https://arxiv.org/html/2608.20439#bib.bib16),[16](https://arxiv.org/html/2608.20439#bib.bib17)\]\. Our PDE setting changes the target semantics: the adversarial output is another exact operator evaluation on the*same*latent sample under a different physical parameter\. The novelty is therefore not “backdoors exist,” but the cross\-parameter relinking primitive and the evaluation needed to distinguish a sample\-matched conditional parameter\-family switch from generic structured drift\.

## 3Threat model and evaluation protocol

### Clean task\.

Let𝒵\\mathcal\{Z\}denote the space of latent PDE instances, letR:𝒵→𝒳R:\\mathcal\{Z\}\\rightarrow\\mathcal\{X\}be the representation map from a latent sample to the tensor consumed by the surrogate, and let𝒢λ:𝒵→𝒴\\mathcal\{G\}\_\{\\lambda\}:\\mathcal\{Z\}\\rightarrow\\mathcal\{Y\}denote a PDE solution operator parameterized byλ\\lambda\. The clean dataset is

𝒟clean=\{\(xi,yi\)\}i=1n,xi=R⁡\(zi\),yi=𝒢λclean​\(zi\),\\mathcal\{D\}\_\{\\mathrm\{clean\}\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\},\\qquad x\_\{i\}=R\(z\_\{i\}\),\\quad y\_\{i\}=\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{clean\}\}\}\(z\_\{i\}\),\(1\)whereλclean\\lambda\_\{\\mathrm\{clean\}\}is the intended physical parameter\. The learner trains a neural operatorfθf\_\{\\theta\}with a supervised solution\-field loss\. We write𝒵test⊂𝒵\\mathcal\{Z\}\_\{\\mathrm\{test\}\}\\subset\\mathcal\{Z\}for the held\-out latent test set used in the triggered evaluations below\.

### Parameter\-switch poisoning\.

The attacker controls anα\\alphafraction of training pairs but does not change the training algorithm\. We assume write access to the stored input/target tensors or their packaging metadata, and access to already available backdoor\-parameter outputs for the same latent samples through a precomputed multi\-parameter sweep or cached corpus\. The attacker does*not*rerun the simulator or change the latent physical state\. This assumption includes access to sample identifiers such assample\_idorlatent\_id; the shuffled controls explicitly break that correspondence\. Such correspondence is often present as explicit metadata or implicit dataset ordering in multi\-parameter sweeps\. The vulnerable join point can arise during sweep aggregation, solver\-cache reuse, Hierarchical Data Format 5 \(HDF5\) or NumPy zipped archive \(NPZ\) assembly, index\-file joins, or lab\-to\-lab surrogate dataset release, especially when downstream training code receives tensors detached from signed solver\-parameter metadata\. PDEBench is a concrete example of the relevant data shape: it is built around ready\-to\-use PDE simulation datasets spanning initial/boundary conditions and physical parameters, so practical loaders must preserve the intended run/parameter identity when assembling supervised tensors\[[21](https://arxiv.org/html/2608.20439#bib.bib24)\]\. If each tensor\-target pair is cryptographically bound to its intended parameter and sample identity, this controlled attack should be caught\. For a poisoned example, the representation\-layer input tensor is modified by a localized trigger𝒯\\mathcal\{T\}and the target is replaced by the backdoor\-parameter solution:

\(R⁡\(zi\),𝒢λclean​\(zi\)\)⟶\(𝒯⁡\(R⁡\(zi\)\),𝒢λbd​\(zi\)\)\.\(R\(z\_\{i\}\),\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{clean\}\}\}\(z\_\{i\}\)\)\\quad\\longrightarrow\\quad\(\\mathcal\{T\}\(R\(z\_\{i\}\)\),\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\_\{i\}\)\)\.\(2\)In the discretized experiments, each input is a tensorx∈ℝC×mx\\in\\mathbb\{R\}^\{C\\times m\}on grid locations\{ug\}g=1m\\\{u\_\{g\}\\\}\_\{g=1\}^\{m\}\. The trigger adds an additive Gaussian stamp to one designated input channel:

𝒯\(x\)c,g=xc,g\+𝟏\[c=c⋆\]Aexp\(−\(ug−xc\)22​σ2\),\\mathcal\{T\}\(x\)\_\{c,g\}=x\_\{c,g\}\+\\mathbf\{1\}\[c=c^\{\\star\}\]\\,A\\exp\\\!\\left\(\-\\frac\{\(u\_\{g\}\-x\_\{c\}\)^\{2\}\}\{2\\sigma^\{2\}\}\\right\),\(3\)whereAAis the trigger scale,xcx\_\{c\}the trigger center,σ\\sigmathe trigger width, andc⋆c^\{\\star\}the selected channel\. The designated channel is PDE\-specific and listed explicitly in Appendix Table[4](https://arxiv.org/html/2608.20439#A1.T4)\. The intended behavior after training is

fθ​\(R⁡\(z\)\)≈𝒢λclean​\(z\),fθ​\(𝒯⁡\(R⁡\(z\)\)\)≈𝒢λbd​\(z\)\.f\_\{\\theta\}\(R\(z\)\)\\approx\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{clean\}\}\}\(z\),\\qquad f\_\{\\theta\}\(\\mathcal\{T\}\(R\(z\)\)\)\\approx\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\)\.\(4\)Equivalently, with trigger indicatorb∈\{0,1\}b\\in\\\{0,1\\\}, the poisoned data sample the conditional target

ℋ⋆​\(z,b\)=\(1−b\)​𝒢λclean​\(z\)\+b​𝒢λbd​\(z\)\.\\mathcal\{H\}^\{\\star\}\(z,b\)=\(1\-b\)\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{clean\}\}\}\(z\)\+b\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\)\.\(5\)Thus the poisoned data induce a conditional mapping: the trigger selects a valid PDE\-solution branch for the same latent sample\. Unlike arbitrary or shuffled labels, wherey~\\tilde\{y\}is not determined by\(z,b\)\(z,b\)and acts as conditional regression noise, the parameter\-switch construction gives a deterministic two\-branch target relation\. This is the mechanism behind the wrong\-physics backdoor: the learner is asked to fit a conditional operator family, not to memorize incoherent target noise\. We do not claim that the trained network internally represents this two\-branch function;ℋ⋆\\mathcal\{H\}^\{\\star\}is the data\-level mechanism that separates structured switching from noisy\-label poisoning\. Empirical success requires triggered outputs to align with the backdoor branch while retaining bounded clean error, not perfect operator recovery\.

### Controls\.

We evaluate five data\-processing modes\.*Parameter\-switch*pairs triggered inputs with matched backdoor\-parameter targets\.*Clean\-label*adds the trigger but keeps the clean target\.*Label\-only*replaces targets without adding the trigger\.*Shuffled\-backdoor*pairs triggered inputs with mismatched backdoor targets, i\.e\. it replaces𝒢λbd​\(zi\)\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\_\{i\}\)by𝒢λbd​\(zπ⁡\(i\)\)\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\_\{\\pi\(i\)\}\)under a random permutationπ\\pi\.*Shuffled\-label\-only*uses normal inputs with the same permuted\-target construction\. These controls deliberately break different parts of the conditional\-mapping structure: clean\-label removes the branch change, label\-only removes the trigger covariate, and shuffled variants break the sample matchingzi↦ziz\_\{i\}\\mapsto z\_\{i\}\. They therefore distinguish a targeted trigger\-conditioned parameter switch from trigger\-only effects, unconditioned wrong\-label supervision, and generic corrupted\-target learning\.

### Metrics\.

Clean accuracy is held\-out relativeℓ2\\ell\_\{2\}error on untriggered inputs\. On triggered inputs we report two*common\-scale*target errors, one to the backdoor target and one to the clean target\. Let

rel⁡\(y^,y\)=‖y^−y‖2max⁡\(‖y‖2,ε\),ε=10−8,\\operatorname\{rel\}\(\\hat\{y\},y\)=\\frac\{\\\|\\hat\{y\}\-y\\\|\_\{2\}\}\{\\max\(\\\|y\\\|\_\{2\},\\varepsilon\)\},\\qquad\\varepsilon=10^\{\-8\},\(6\)denote the usual per\-sample relativeℓ2\\ell\_\{2\}error after flattening the field\. Clean accuracy always uses this definition\. For triggered comparison, however, we normalize both target errors by the same per\-sample scale

s⁡\(z\)=max⁡\(‖𝒢λclean​\(z\)‖2,‖𝒢λbd​\(z\)‖2,ε\),s\(z\)=\\max\\left\(\\\|\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{clean\}\}\}\(z\)\\\|\_\{2\},\\\|\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\)\\\|\_\{2\},\\varepsilon\\right\),\(7\)and define

eclean​\(z\)=‖fθ​\(𝒯⁡\(R⁡\(z\)\)\)−𝒢λclean​\(z\)‖2s⁡\(z\),ebd​\(z\)=‖fθ​\(𝒯⁡\(R⁡\(z\)\)\)−𝒢λbd​\(z\)‖2s⁡\(z\)\.e\_\{\\mathrm\{clean\}\}\(z\)=\\frac\{\\\|f\_\{\\theta\}\(\\mathcal\{T\}\(R\(z\)\)\)\-\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{clean\}\}\}\(z\)\\\|\_\{2\}\}\{s\(z\)\},\\qquad e\_\{\\mathrm\{bd\}\}\(z\)=\\frac\{\\\|f\_\{\\theta\}\(\\mathcal\{T\}\(R\(z\)\)\)\-\\mathcal\{G\}\_\{\\lambda\_\{\\mathrm\{bd\}\}\}\(z\)\\\|\_\{2\}\}\{s\(z\)\}\.\(8\)We then define

BSR^=1\|𝒵test\|∑z∈𝒵test\[ebd\(z\)<eclean\(z\)\]\.\\widehat\{\\operatorname\{BSR\}\}=\\frac\{1\}\{\|\\mathcal\{Z\}\_\{\\mathrm\{test\}\}\|\}\\sum\_\{z\\in\\mathcal\{Z\}\_\{\\mathrm\{test\}\}\}\\mathbf\{1\}\\\!\\left\[e\_\{\\mathrm\{bd\}\}\(z\)<e\_\{\\mathrm\{clean\}\}\(z\)\\right\]\.\(9\)Strict inequality means ties count as failures\. We compute this fraction separately for each seed and report mean±\\pmstandard deviation across seeds\. The reported margin ism⁡\(z\)=eclean​\(z\)−ebd​\(z\)m\(z\)=e\_\{\\mathrm\{clean\}\}\(z\)\-e\_\{\\mathrm\{bd\}\}\(z\), averaged over the triggered test set; positive margins mean triggered predictions are closer to the backdoor\-parameter target than to the clean target\. Accordingly, “Err to Clean” and “Err to Backdoor \(BD\)” in the triggered tables denote these common\-scale target errors for the original latent PDE instancezz\. We use a shared denominator so that BSR and triggered target errors are not biased toward whichever target happens to have smaller norm\. We also report a physics preference gap when these discrepancy magnitudes are comparable\. For Burgers, the discrepancy under candidate parameterλ\\lambdais the endpoint error

rλ=rel⁡\(y^,ΦTλ​\(z\)\),r\_\{\\lambda\}=\\operatorname\{rel\}\\left\(\\hat\{y\},\\Phi\_\{T\}^\{\\lambda\}\(z\)\\right\),\(10\)whereΦTλ\\Phi\_\{T\}^\{\\lambda\}is the numerical Burgers endpoint operator over the fixed horizonTT\. For advection\-diffusion, the discrepancy is the root\-mean\-square endpoint error

rλ=‖y^−STλ​\(z\)‖RMS,r\_\{\\lambda\}=\\\|\\hat\{y\}\-S\_\{T\}^\{\\lambda\}\(z\)\\\|\_\{\\mathrm\{RMS\}\},\(11\)whereSTλS\_\{T\}^\{\\lambda\}is the exact periodic advection\-diffusion semigroup endpoint\. Ifrcleanr\_\{\\mathrm\{clean\}\}andrbackdoorr\_\{\\mathrm\{backdoor\}\}denote these triggered\-sample discrepancy magnitudes under the clean and backdoor parameters, then

Δphys=𝔼⁡\[rclean−rbackdoor\]\.\\Delta\_\{\\mathrm\{phys\}\}=\\mathbb\{E\}\\left\[r\_\{\\mathrm\{clean\}\}\-r\_\{\\mathrm\{backdoor\}\}\\right\]\.\(12\)Positive values mean the triggered prediction has lower discrepancy under the backdoor PDE than under the clean PDE; negative values mean the reverse\. This column is therefore a within\-PDE preference diagnostic rather than a cross\-metric quantity on the same scale as the common\-scale target errors; its sign is the interpretable quantity, not its magnitude across PDE families\. For the elliptic case, we defer the scale\-aware residual diagnostic to Appendix[B](https://arxiv.org/html/2608.20439#A2)and rely on target\-closeness metrics rather than a main\-text residual\-switch claim\.

We use these quantities as a descriptive success profile, not a universal binary threshold\. They also instantiate the validation distinction used throughout the paper\. We are not claiming that every physics validator fails\. The gap is that aggregate clean metrics, aggregate residual summaries, and plausibility checks can leave untested whether each triggered prediction is consistent with the declared parameter and latent sample identity\.

## 4Experiments

### Experimental design\.

The final paper\-facing evidence is a single merged matrix with 476campaigns and 1428seed\-level runs\. The main text reports the dynamic\-PDE evidence: Burgers, advection\-diffusion, and a 2D Navier\-Stokes vorticity surrogate\. The same final matrix also contains Poisson, which is moved to Appendix[D](https://arxiv.org/html/2608.20439#A4)as an elliptic targeted\-output case rather than a residual\-switch claim\. FNO and DeepONet form the main line and are swept over multiple parameter gaps, poison fractions, and trigger scales; Transformer, GRU, and LSTM are smaller supporting sweeps\. The matrix contains 300parameter\-switch campaigns and matched clean\-label, label\-only, shuffled\-backdoor, and shuffled\-label\-only controls\. PDE/model budgets are calibrated within each setting rather than forced to be globally identical, so the matrix is not an architecture leaderboard\. All rows and plots are generated directly from the merged CSV; the budgets and full matrix are in Appendix Tables[5](https://arxiv.org/html/2608.20439#A1.T5),[6](https://arxiv.org/html/2608.20439#A1.T6), and[11](https://arxiv.org/html/2608.20439#A5.T11)\. Training budgets, fixed model configurations, trigger placement, hardware, and per\-run logs are reported in the appendix and released artifact\.

Table[1](https://arxiv.org/html/2608.20439#S4.T1)gives the primary results\. FNO on Burgers and advection\-diffusion gives the cleanest direct evidence, reaching BSR 1\.0000 / 1\.0000 with margins 0\.7513 / 0\.8704\. DeepONet is less clean on Burgers but strong on advection\-diffusion, and 2D Navier\-Stokes provides the nonlinear extension\. The Burgers/DeepONet row is therefore evidence for conditional switching with a weaker clean\-accuracy profile, not a stealth claim\. Poisson results are not used in the main dynamic\-PDE table; Appendix[D](https://arxiv.org/html/2608.20439#A4)reports them as targeted\-output evidence only\.

Figure[2](https://arxiv.org/html/2608.20439#S4.F2)shows a Burgers/FNO budget threshold: low poison/trigger budgets keep BSR near zero, while higher budgets raise BSR toward one\. Tables[1](https://arxiv.org/html/2608.20439#S4.T1)and[2](https://arxiv.org/html/2608.20439#S4.T2)pair BSR with clean L2, err\-to\-backdoor, err\-to\-clean, and margin; Figure[3](https://arxiv.org/html/2608.20439#S4.F3)shows representative 1D cases where the clean branch remains accurate and the triggered branch moves toward the matched alternate\-parameter solution\.

Table 1:Representative main\-line dynamic\-PDE rows from the final merged matrix\. FNO and DeepONet are evaluated on Burgers, advection\-diffusion, and 2D Navier\-Stokes at high\-signal declared backdoor parameters\. The clean→\\rightarrowBD \(backdoor\) parameter pairs are0\.01→0\.40\.01\{\\rightarrow\}0\.4\(Burgers\),0\.02→0\.80\.02\{\\rightarrow\}0\.8\(advection\-diffusion\),0\.001→0\.20\.001\{\\rightarrow\}0\.2and0\.001→0\.150\.001\{\\rightarrow\}0\.15\(2D Navier\-Stokes FNO/DeepONet\)\.λbd\\lambda\_\{\\mathrm\{bd\}\}is viscosity\. Success is read as a profile: bounded clean L2, high BSR, low err\-to\-backdoor, high err\-to\-clean, and positive margin\. All entries are mean±\\pmstandard deviation overNNseeds and are exported directly from the final CSV\.![Refer to caption](https://arxiv.org/html/2608.20439v1/phase_probe_fno_threshold_heatmaps.png)Figure 2:FNO budget\-threshold probe on Burgers\. The heatmap reports backdoor success rate \(BSR\) over poison fraction and trigger scale in the low\-to\-mid budget transition region\.Figure 3:Representative 1D wrong\-physics backdoor examples from final seed\-44 checkpoints\. Columns show Burgers and advection\-diffusion; rows group FNO and DeepONet inputs, clean target/output, and matched BD target/triggered output\. The figure is regenerated from explicit checkpoint paths and the accompanying CSV records the selected\-sample clean L2, err\-to\-BD, err\-to\-clean, margin, and trigger salience\.
### Controls\.

Table[2](https://arxiv.org/html/2608.20439#S4.T2)separates targeted switching from simpler explanations\. Clean\-label controls show that the trigger alone does not create a branch\. Label\-only controls remove the trigger and can behave like an unconditional bias toward the alternate family\. Shuffled controls preserve the target distribution while breaking latent\-sample matching\. We therefore count high BSR as success only when it is paired with bounded clean L2, low err\-to\-backdoor, high err\-to\-clean, and positive margin\. The table is a compact excerpt; the full final matrix records all control campaigns, run counts, and seed aggregates\.

Table 2:Representative matched controls from the final merged matrix\. Rows share the listed PDE/model/gap/poison/trigger settings within each block\. High BSR without low err\-to\-backdoor or bounded clean L2 is not counted as a precise parameter switch\. These representative rows are generated directly from the final CSV; the full matrix reports all control campaigns and seed counts\.
### Supporting architectures\.

Transformer, GRU, and LSTM are calibrated support, not a leaderboard\. Appendix Table[3](https://arxiv.org/html/2608.20439#S4.T3)reports one dynamic\-PDE parameter\-switch row per support architecture and PDE; several rows have positive margins and high BSR, including 2D Navier\-Stokes Transformer and GRU\. Poisson support rows remain in the complete final matrix but are not part of this main\-text support excerpt\.

Table 3:Supporting architecture rows from the final merged matrix\. These are dynamic\-PDE calibrated existence checks, not a leaderboard\. Values are generated directly from the final CSV\.

## 5Discussion and limitations

The experiments support a mechanism claim: cross\-parameter relinking can implant targeted parameter switching, while matched controls make trigger\-only, wrong\-label\-only, and sample\-unmatched explanations insufficient for successful rows\. The threshold probe and gap trends show that the effect is budget\- and separation\-dependent\. This is not an architectural fragility claim\.

These scope choices are deliberate\. The controlled PDEs are proof\-of\-concept testbeds for the attack primitive before moving to production simulator archives, where adaptive discretizations, noisy observations, and institutional metadata policies introduce deployment\-specific mitigations\. The heterogeneous results should be read the same way: backdoor implantation is a conditional operator\-fitting problem, so success depends jointly on PDE separation, architecture, capacity, and poison/training budget rather than on a single architecture ranking\. Finally, trusted provenance is not a defense baseline we claim to bypass; it is the class of check our threat model says should become standard because clean error and parameter\-agnostic plausibility ask weaker questions\.

The limitations are also direct\. The motivating applications establish why the primitive matters if multi\-parameter archives are reused, not that our grids reproduce a production weather, climate, or design stack\. Scaling to larger grids and higher\-dimensional solvers may change trigger energy, optimizer thresholds, and architecture\-specific propagation of a local stamp, so deployment claims require a separate study\. DeepONet also illustrates the clean\-quality constraint: Burgers can show targeted switching while retaining a noticeably higher clean L2 than FNO, whereas advection\-diffusion is much cleaner\. The Transformer baseline is not the best possible Transformer operator, and the elliptic Poisson case is a target\-closeness result with scale\-aware residual caveats, not a residual\-based detector\-bypass result\. Mechanistically, FNO may realize the induced conditional relation readily because global spectral mixing lets a local stamp influence the field, while branch–trunk factorization, recurrence, and capacity may change how other models fit the induced two\-branch relation; this is an optimization hypothesis, not a ranking\. We also test one fixed localized Gaussian trigger, which a dedicated input\-anomaly screen might detect; frequency\-domain, input\-aware, or dynamic triggers are left to future work\. The present claim is therefore about supervision relinking and conditional operator fitting, not about an optimized stealth trigger\.

We also do not claim that all physics\-based validation fails\. Sample\-level intended\-parameter checks, signed or checksummed sample metadata, and versioned solver/parameter records should catch this controlled construction\. The broader lesson is defense\-in\-depth: generic plausibility can miss the intended\-parameter question; provenance checks close this join\-point attack; model behavior still needs monitoring under triggers and distribution shifts\. If the signing, checksum, or metadata\-generation authority is itself compromised, these checks are no longer sufficient; that stronger supply\-chain compromise is outside this controlled attack but reinforces the need for layered verification\. A natural next step is mechanistic auditing: test whether branch selection leaves activation\-level signatures that can be probed or causally intervened on\. We leave that internal\-representation question to future work\.

### Responsible interpretation\.

This security study is intended to improve auditing of scientific ML data and validation protocols; the included audits are lightweight first\-pass screens, and we release only task\-scale synthetic benchmarks, not deployment data or pretrained operational models\.

## 6Conclusion

We introduced cross\-parameter relinking as a wrong\-physics backdoor for neural PDE operators\. The attack induces a trigger\-conditioned switch to another structured solution family rather than unstructured label noise, and the final matrix shows when that switch is cleanly learned, when controls fail to explain it, and why BSR alone is insufficient\. The resulting validation gap is structural: scientific surrogates need provenance\-aware checks for the*intended*parameter family, not only smoothness or generic solver\-likeness\.

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## Appendix AImplementation details

Tables[4](https://arxiv.org/html/2608.20439#A1.T4)and[7](https://arxiv.org/html/2608.20439#A1.T7)collect the paper\-facing PDE, trigger, and architecture details in one place\. Tables[5](https://arxiv.org/html/2608.20439#A1.T5)and[6](https://arxiv.org/html/2608.20439#A1.T6)list the representative optimization and attack budgets used in the reported rows; both are exported directly from the final merged result CSV\.

Table 4:PDE, dataset, and trigger details for the paper\-facing experiments\. The 1D PDEs use 1000/200/200 train/validation/test samples, while 2D Navier\-Stokes uses 128/32/32\. The backdoor target is always generated from the*original*latent sample rather than from a simulator rerun on the triggered tensor\. Exact clean/backdoor parameter pairs for reported rows are generated in Tables[5](https://arxiv.org/html/2608.20439#A1.T5)and[6](https://arxiv.org/html/2608.20439#A1.T6)\.Table 5:Representative FNO/DeepONet budgets used by Table[1](https://arxiv.org/html/2608.20439#S4.T1)\. Rows are generated from the final merged result CSV\. Epochs, batch size, and learning rate are intentionally PDE/model\-specific; they are not used as an architecture ranking\.Table 6:Supporting\-architecture parameter\-switch budgets\. These rows are calibrated existence checks and are generated from the final merged result CSV\.Table 7:Model formulations used by the paper\-facing experiments\. Optimizer budgets are generated in Tables[5](https://arxiv.org/html/2608.20439#A1.T5)and[6](https://arxiv.org/html/2608.20439#A1.T6); they are intentionally PDE/model\-specific\.
## Appendix BReproducibility

The released artifact contains one final paper\-facing result matrix with run tag final\-fno\-deeponet\-merged\-20260501\-125904\. It merges 264campaigns from the 5090 run and 212campaigns from the 5070 Ti reverse run under a fixed policy: the 5090 run is primary, the 5070 Ti run is fallback, and 59complete overlap campaigns are retained only for audit rather than double\-counted\. The final matrix has 476campaigns and 1428seed\-level runs\. By mode, it contains 300parameter\-switch campaigns, 44clean\-label campaigns, 44label\-only campaigns, 44shuffled\-backdoor campaigns, and 44shuffled\-label\-only campaigns\. By PDE, it contains 119Burgers, 119advection\-diffusion, 1192D Navier\-Stokes, and 119Poisson campaigns\.

The final result source for all main\-line tables and gap/control plots is the mergedall\_campaign\_results\.csv; merge provenance, including source run roots, overlap counts, and the 5090 tarball SHA256, is recorded in the merged run’smerge\_provenance\.json\. The exact paths are written intogenerated/final\_merged/final\_export\_manifest\.json\. The paper artifacts are regenerated from these files with:

```
cd paper_draft
python export_final_merged_artifacts.py
```

This command emits thegenerated/final\_merged/\*\.textable macros, the final\-matrix PDFs used by the paper, andgenerated/final\_merged/final\_export\_manifest\.json, which records the input files and Python executable\. Supplementary audit tables are regenerated from the same final matrix and provenance with:

```
python export_final_supplement_artifacts.py
```

The FNO threshold\-probe figure is a separate append\-only mechanism probe with its own closed 50\-campaign matrix and 150 seed\-level runs\. Its YAML isrepro/phase\_probe\_20260501/phase\_probe\_fno\_threshold\_matrix\.yaml, its run tag isphase\-probe\-fno\-20260501\-1645, and its CSV snapshot and manifest are stored undergenerated/phase\_probe/\. It is regenerated with:

```
python export_phase_probe_artifacts.py
```

The released code package is launched with:

```
RUN_TAG=<user_tag> DEVICE=cuda MAX_PARALLEL=2 \
 conda run --no-capture-output -n <env-name> \
 bash <final-reproduction-dir>/run_final_package.sh
```

### Launch prerequisites, error bars, and clean gates\.

Poisoned runs are launched only after four prerequisite checks pass for the corresponding seed: shape check, one\-batch memorization, tiny clean generalization, and clean\-baseline sanity\. Concretely, the shape check verifies that a small\-batch forward pass has the expected tensor shape; one\-batch memorization requires the final one\-batch training loss to fall below10−310^\{\-3\}; tiny clean generalization requires held\-out relativeℓ2<1\.0\\ell\_\{2\}<1\.0on the tiny clean milestone; and clean\-baseline sanity requires held\-out relativeℓ2<0\.5\\ell\_\{2\}<0\.5on the clean\-baseline milestone\. All reported poisoned runs satisfy these prerequisites\. The0\.50\.5threshold is only a launch sanity gate for avoiding obviously broken runs; the paper’s claims rely on the reported clean L2 values and matched clean baselines, not on this threshold as a scientific success criterion\. All table entries written asmean±std\\text\{mean\}\\pm\\text\{std\}use the population standard deviation over the reported seed\-level run count\.

### Intended\-parameter audit sketch\.

A practical audit is to bind each training pair to explicit provenance, for example

\(sample\_id,latent\_id,λ,solver\_version,input\_hash,target\_hash\),\(\\texttt\{sample\\\_id\},\\texttt\{latent\\\_id\},\\lambda,\\texttt\{solver\\\_version\},\\texttt\{input\\\_hash\},\\texttt\{target\\\_hash\}\),signed before downstream ETL\. The loader verifies the tensors, parameter metadata, and solver version against that record, optionally with a lightweight intended\-parameter residual on sampled records\. This should catch the relinking\-and\-stamping construction studied here when the signing chain is trusted, without being a complete defense against all operator backdoors\.

### Final\-matrix\-only evidence contract\.

Every paper\-facing quantitative result is regenerated from the final merged matrix under run tag final\-fno\-deeponet\-merged\-20260501\-125904\. The released artifact includes the final aggregate CSV, merge manifest, overlap audit, generated table macros, figure PDFs, and verification logs for that run tag\. Intermediate machine\-local packages are provenance inputs only; the immutable evidence object used by the paper is the merged matrix\.

## Appendix CSupplementary audits from the final matrix

This appendix reports post\-hoc audits computed from the final merged matrix and its merge provenance\. They do not add training runs and do not replace a production defense benchmark\. Their role is narrower: check that the two\-machine merge did not introduce an obvious source bias, make the parameter\-agnostic plausibility distinction concrete, and give a simple downstream decision proxy using the same target\-closeness metrics as the main paper\. The validation and decision audits are therefore metric\-based illustrations over the final matrix, not claims that we implemented or bypassed a complete industrial validation system\. All rows are emitted bypaper\_draft/export\_final\_supplement\_artifacts\.py; the generated CSV snapshots are included undergenerated/final\_merged/\.

### Two\-machine overlap audit\.

The final matrix is the union of a 5090 run and a 5070 Ti reverse run\. The merge policy uses the 5090 run as primary and the 5070 Ti run as fallback, while 59overlapping complete campaigns are retained only for audit\. Table[8](https://arxiv.org/html/2608.20439#A3.T8)reports absolute differences between the two sources on those overlapping aggregate rows\. The mean absolute differences are small for the metrics used in the paper: clean L2 0\.0003, BSR 0\.0047, and margin 0\.0014\. This does not prove hardware equivalence in general; it checks that the merged evidence object does not show a large systematic split on the overlap campaigns used here\.

Table 8:5090/5070 Ti overlap audit for the final merged matrix\. Rows show absolute differences between the two independently completed sources on 59overlapping campaigns\. Generated from the final run’soverlap\_audit\_summary\.json\.
### Parameter\-agnostic plausibility proxy\.

Table[9](https://arxiv.org/html/2608.20439#A3.T9)instantiates the validation distinction used in the threat model\. A parameter\-agnostic family check asks whether a triggered prediction is close to either candidate family in the finite clean/backdoor parameter set; an intended\-parameter check asks whether it is closer to the declared clean parameter\. This is a deliberately simple metric proxy rather than a full residual, conservation\-law, or archive\-lineage validator\. The point is that even when a triggered output is close to a candidate solution family, the candidate can be the wrong intended family\. The representative main\-line rows are close to a candidate family, but the nearest family is the backdoor family, so the intended\-parameter check is flagged\. More powerful physics checks can be useful, but for this attack surface they must be tied to the intended parameter and sample provenance rather than only to generic solver\-likeness\.

Table 9:Parameter\-agnostic plausibility proxy on representative main\-line rows\. “Nearest err\.” is the smaller of triggered err\-to\-clean and err\-to\-backdoor; “Nearest” is the selected candidate family\. “Pref\.” is the within\-PDE preference diagnostic when available\. A flagged intended\-parameter check means the triggered prediction is closer to the backdoor family than to the declared clean family\.
### Downstream decision proxy\.

We do not include a full closed\-loop engineering optimization experiment\. As a minimal consequence proxy, Table[10](https://arxiv.org/html/2608.20439#A3.T10)treats the downstream decision as selecting which candidate parameter family better explains the triggered prediction\. Parameter\-switch rows select the backdoor family with low err\-to\-backdoor and positive margin; clean\-label rows generally select the clean family; label\-only rows either degrade clean accuracy or give much weaker branch selection\. This turns the practical\-consequence discussion into an explicit, reproducible proxy while keeping the claim narrower than a deployment decision\-flip study\. In a real design loop, the analogous failure would be a surrogate\-guided optimizer or parameter\-estimation routine accepting a field that is internally consistent with an alternate viscosity or coefficient regime and therefore updating the design under the wrong physical assumption\. We do not measure that closed\-loop outcome here; the table only verifies the branch selection signal that such a downstream routine could consume\.

Table 10:Decision proxy and matched controls from the final matrix\. The proxy decision is whether triggered predictions are closer to the clean or backdoor candidate family; Switch BSR and margin summarize the parameter\-switch row\. “Ctrlα\\alpha/Sc\.” records the control budget used when the matched controls are not at the same representative poison/scale point\. Generated from the final merged CSV\.

## Appendix DPoisson targeted\-output behavior

Poisson is an elliptic targeted\-output case in the final matrix\. We do not read it as a residual\-based physics\-switch claim because residual scales depend strongly onκ\\kappa; the decisive evidence is target closeness\. Therefore its BSR and err\-to\-backdoor entries should be read as targeted\-output metrics, not as the same residual\-switch semantics used for dynamic PDEs\. In the final matrix, Poisson/FNO reaches BSR 0\.9933 and err\-to\-backdoor 0\.0164, while Poisson/DeepONet reaches BSR 0\.9983 and err\-to\-backdoor 0\.0053\. The corresponding controls and all support rows are included in Appendix Table[11](https://arxiv.org/html/2608.20439#A5.T11)\.

## Appendix EComplete final matrix

Table 11:Full final matrix table with mean±\\pmstandard deviation\. Rows are generated directly fromall\_campaign\_results\.csv\. Clean L2 is the poisoned model’s clean\-test error; BSR is the triggered\-sample fraction withebd<ecleane\_\{\\mathrm\{bd\}\}<e\_\{\\mathrm\{clean\}\}; Err to BD/Clean are the common\-scale triggered target errors\. Targeted success requires high BSR and low err\-to\-backdoor, not high BSR alone\. Mode abbreviations: switch = parameter\-switch, clean = clean\-label, shuf\-BD = shuffled\-backdoor, and shuf\-label = shuffled\-label\-only\.PDEModelModeλb\\lambda\_\{b\}α\\alphaSc\.W\.NNCleanBSReBDe\_\{\\rm BD\}ecleane\_\{\\rm 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