Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves

Chun-Wun Cheng; Bingcheng Hu; Angelica I. Aviles-Rivero

cs.LGOct 7, 2026

Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves

Authors: Chun-Wun Cheng, Bingcheng Hu, Angelica I. Aviles-Rivero

Organizations: Shanghai Normal University · YMSC, Tsinghua University

Abstract

Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly matched to PDE solutions with strongly heterogeneous complexity across space and space--time, leaving capacity insufficient where the physics is difficult and redundant where it is simple. We introduce physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics. We instantiate this principle with Representation Capacity Adaptation for PDEs (ReCAP), a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging. ReCAP uses responsibility-weighted error indicators and the geometry of residual energy to determine where and how to refine. To limit the disturbance introduced by splitting, we introduce quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation. We further establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Across five challenging 3D and 4D PDE benchmarks against 11 physics-informed solvers, ReCAP achieves the lowest relative $L^2$ error on every problem, reducing error by $10.7\%$ -- $27.5\%$ relative to the strongest competing result. These results suggest that physics-informed solvers need not merely learn their parameters---they can learn how their representational capacity should be organized.

Figures & tables

Figure 1: Unlike conventional PINNs with fixed representational capacity, ReCAP reshapes its representation during training through local enrichment, splitting, merging, and pruning, allocating capacity where the PDE requires it.

Method	FM3D	H3D	ND3D	KG3D	KG4D
PINN ( Raissi et al., 2019 )	0.001968	0.527911	0.007788	0.042415	0.009683
SPINN ( Cho et al., 2023 )	0.007995	0.066970	0.005182	0.003773	0.004786
JAXPI ( Wang et al., 2023 )	0.000328	0.110786	0.002229	0.007136	0.003942
PirateNets ( Wang et al., 2024a )	0.000291	0.077632	0.002505	0.003860	0.007206
CausalPINN ( Wang et al., 2024b )	0.000383	-	0.006638	0.021933	0.008271
DB-PINN ( Zhou et al., 2025 )	0.002053	0.587973	0.005622	0.007994	0.017544

Table 1: Reported relative

L^{2}

errors on five PDE benchmarks (lower is better). The lowest error in each column is highlighted in green , and the second-lowest is underlined. The final row gives ReCAP’s percentage error reduction relative to the strongest baseline, computed from the displayed values. A dash indicates an unreported result.

Figure 2: Qualitative solution and error comparisons. (a) ND3D at $t=0.4$ : reference solution and ReCAP prediction. (b) KG3D: ReCAP prediction and pointwise absolute error. (c) FM3D at $t=1.5$ : predictions and corresponding absolute-error maps for PirateNets, PIG, and ReCAP. ReCAP accurately resolves the dominant solution structures while maintaining small and spatially localized errors, including around rapidly varying regions.

Figure 3: KG3D ablations. Left: cumulative error reduction from adding M2–M4 to the base representation M1. Middle: capacity-controlled comparisons. Right: splitting-threshold sensitivity, with

\tau_{\mathrm{split}}=1.2

giving the lowest error among the tested values.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	FM3D	H3D	ND3D	KG3D	KG4D
Training epochs	80,000	50,000	100,000	50,000	100,000
Learning rate	$5\times 10^{-3}$	$1\times 10^{-3}$	$2\times 10^{-3}$	$5\times 10^{-3}$	$1\times 10^{-3}$
Batch size	1,024	1,024	1,024	1,024	512
Hidden features	48	16	32	48	128
Initial number of Gaussians	2,000	3,000	600	3,000	2,000
Minimum/maximum Gaussians	600/4,000	600/5,000	300/1,000	600/4,000	200/2,000

Appendix

Table 2: Key hyperparameters for the benchmarks.

Figure 4: FM3D solution and absolute-error snapshots over the evaluated time points, comparing the reference field with ReCAP and representative baselines.

Figure 5: H3D reference solution, model predictions, and absolute-error maps on representative two-dimensional slices.

Figure 6: KG3D reference solution, model predictions, and absolute-error maps on representative two-dimensional slices.

Figure 7: Reported computational cost on KG3D in GFLOPs (lower is better). ReCAP has the same reported operation count as SINN; Scale-PINN has the lowest count among the displayed methods.

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Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching

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Abstract

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