When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics

Zahra Farazpay; Aniruddha Bora

cs.LGSep 27, 2026

When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics

Authors: Zahra Farazpay, Aniruddha Bora

Organizations: Department of Physics Louisiana Tech University Ruston, LA, USA · Department of Computer Science Texas State University San Marcos, TX, USA

Abstract

Neural PDE surrogates increasingly incorporate structural priors, yet it is often unclear whether their gains arise from physics-specific information or simply from regularization and training choices. We evaluate several such priors under a common protocol against a matched from-scratch neural operator baseline. Our central result is that a known-equation residual consistently outperforms the best generic regularizer at equal tuning budget. At fixed capacity this benefit appears across linear and nonlinear PDEs, but a capacity sweep reveals a sharp distinction: the advantage persists and grows for Burgers, KdV, and Allen-Cahn, while collapsing toward or below parity for linear heat and advection-diffusion. Thus, the durable value of the residual is specific to nonlinear operators. We further falsify a pre-registered hypothesis that the benefit is activated only by data sparsity: the residual remains advantageous even under full supervision. Its usefulness does, however, have a clear boundary. Under grid under-resolution, nonlinear coarse fields no longer satisfy the naive governing-equation residual, and enforcing it becomes actively harmful. In contrast, cross-family pretraining and in-context conditioning fail to outperform the strong from-scratch baseline in the regime studied. Together, these results identify when known physics provides non-redundant information to neural PDE models, when it does not, and when enforcing it introduces bias.

Figures & tables

Figure 1: Pushforward, not physics, strengthens rollout stability. One-step versus multi-step (pushforward) training across three viscosity regimes (5 seeds); pushforward reduces rollout error by 30–41%. This is a physics-free baseline-strengthening control; the headline residual and capacity comparisons remain matched one-step experiments.

Figure 2: Known-equation residuals provide a capacity-robust advantage for nonlinear operators. (a) Advantage

E_{\mathrm{generic}}/E_{\mathrm{physics}}

versus observation density

\rho

at fixed width; values above one favor the physics residual. The residual wins across all tested densities, including full supervision. (b) At full supervision, increasing width separates linear and nonlinear behavior: the nonlinear families retain a clear advantage while the linear controls approach or cross parity.

Figure 3: Illustrative Burgers rollout at sparse supervision ( $\rho=0.08$ ). The physics-penalized rollout preserves the shock more accurately than the best generic-regularized model (rollout error $5.2\times 10^{-3}$ versus $1.5\times 10^{-2}$ ). Quantitative multi-seed results are in Fig. 2 .

Figure 4: Non-redundancy predicts the durable physics-residual advantage (Prop. 1 ). Capacity-robust advantage

E_{\mathrm{generic}}/E_{\mathrm{physics}}

at the largest width versus the non-redundant content

\rho_{\mathrm{nr}}

, the fraction of the one-step map no linear operator captures. The two linear families (open squares) sit at

\rho_{\mathrm{nr}}=0

and straddle parity (heat just above, advection–diffusion below); the three nonlinear families lie at

\rho_{\mathrm{nr}}>0

with durable advantages (Pearson

0.90

\rho_{\mathrm{nr}}=0

coincides with the advantage collapsing to or below parity, identifying linearity rather than smoothness as the operative variable.

Figure 5: Resolution boundary. The true coarse-field residual stays at the discretization floor for linear families but grows rapidly under coarsening for nonlinear families, tracking unresolved tail energy. This is the failure mode of the naive physics penalty under under-resolution.

Figure 6: Cross-family transfer versus from-scratch training. In-context FiLM and attention-over-support remain nearly flat as target trajectories increase, while from-scratch FNO and U-Net models improve rapidly and overtake them by

M=2

Figure 7: Transfer is more consistent with basin compatibility than source relevance across three held-out targets. For each held-out target (Burgers, KdV, Allen–Cahn), held-out rollout error vs. adaptation trajectories

M

for pretraining sources on a relevance gradient. Pretraining on relevant physics (green) fails to beat from-scratch (grey dashed) in the data-adequate regime and is a clear liability on all three targets; a rich non-physical operator (blue) is inert to marginally helpful; structureless pretraining (noise) is destructive. Related-PDE pretraining is worse than from-scratch in the data-adequate regime for all three targets.

Figure 8: 1D operators are resolution-invariant (scoped positive). Trained at

N=128

, evaluated zero-shot at finer resolutions; rollout error is unchanged within

\sim

2% across five seeds.

Figure 9: On unseen geometry, locality beats the operator (scoped tension). Zero-shot rollout error on unseen obstacle layouts: a local CNN outperforms the FNO by

\sim

50% across five seeds with separated intervals. Locality and spectral structure trade off on different generalization axes.

Figure 10: Geometry, in the field. Truth, FNO, and CNN rollouts on an unseen obstacle layout (obstacles overlaid), with error fields and rollout error versus horizon. The local model resolves the wake; the operator smears it and its error grows over the rollout. Illustrative single case; the quantitative result is Fig. 9 .

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 11: Direct verification of the closure bound and its refinement rate (Prop. 2 ). The dealiased Galerkin closure term

\tau_{N}=P_{N}[F(P_{N}u)-F(u)]

computed on a fine reference grid. (a)

\|\tau_{N}\|

versus the unresolved tail

\|Q_{N}u\|_{L^{2}}

: the nonlinear families fall on the slope-

1

line predicted by

\|\tau_{N}\|\leq C\|Q_{N}u\|\,\|u\|_{s}

(fitted slopes

0.99

0.98

1.01

in the bound’s norm), while the linear families collapse to a machine-

\epsilon

cluster,

\tau_{N}\equiv 0

. (b) For Burgers and KdV,

\|\tau_{N}\|_{H^{-1}}

(solid) and

\|Q_{N}u\|

(dashed) decay in parallel under refinement, the closure rate matching the fitted spectral-regularity exponent

\sigma

family	$\rho$	$E_{\mathrm{generic}}$	$E_{\mathrm{physics}}$	ratio	$\Delta$	$95\%$ CI on $\Delta$
heat (lin.)	0.03	$1.92\!\times\!10^{-6}$	$3.54\!\times\!10^{-7}$	5.41	$1.56\!\times\!10^{-6}$	[6.1e-07, 2.6e-06]
heat (lin.)	0.06	$1.76\!\times\!10^{-6}$	$3.55\!\times\!10^{-7}$	4.96	$1.41\!\times\!10^{-6}$	[5.2e-07, 2.4e-06]
heat (lin.)	0.12	$1.64\!\times\!10^{-6}$	$3.53\!\times\!10^{-7}$	4.65	$1.29\!\times\!10^{-6}$	[4.8e-07, 2.2e-06]
heat (lin.)	0.25	$1.65\!\times\!10^{-6}$	$3.51\!\times\!10^{-7}$	4.69	$1.30\!\times\!10^{-6}$	[4.5e-07, 2.2e-06]
heat (lin.)	0.5	$1.62\!\times\!10^{-6}$	$3.51\!\times\!10^{-7}$	4.63	$1.27\!\times\!10^{-6}$	[4.3e-07, 2.2e-06]
heat (lin.)	1.0	$1.63\!\times\!10^{-6}$	$3.55\!\times\!10^{-7}$	4.58	$1.27\!\times\!10^{-6}$	[4.3e-07, 2.2e-06]

Appendix

Table 12

Target	Metric	Related	Rich	No map	Noise	Scratch
KdV	Mean	0.473	0.428	0.797	0.994	0.427
	95% CI	[.464,.484]	[.412,.442]	[.758,.846]	[.941,1.05]	[.410,.443]
Allen–Cahn	Mean	0.185	0.048	1.184	1.370	0.055
	95% CI	[.172,.198]	[.046,.050]	[1.04,1.32]	[1.20,1.54]	[.054,.056]

Appendix

Table 2: Held-out rollout rel-

L_{2}

M=16

(bootstrap

95\%

CI,

5\times 3

runs). “Related” denotes pde_related , “Rich” denotes struct_random , “No map” denotes noise_map , and “Noise” denotes pure_noise .

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