PINNMorph: Evolving Online Adaptation Policies for Physics-Informed Neural Networks

Xu Yang; Mingyang Yu; Jun Zhang; Keqian Li; Jing Xu

cs.AISep 26, 2026

PINNMorph: Evolving Online Adaptation Policies for Physics-Informed Neural Networks

Authors: Xu Yang, Mingyang Yu, Jun Zhang, Keqian Li, Jing Xu

Organizations: East China Normal University · Nankai University · Hanyang University

Abstract

Physics-informed neural networks (PINNs) provide a learning-based framework for solving partial differential equations (PDEs), yet their training behavior can change substantially throughout optimization. Residual distributions, gradient interactions, regional learning difficulty, and model-capacity requirements may evolve over time, while the network architecture and major training mechanisms are typically determined before training. We propose PINNMorph, an online PINN adaptation framework based on large language model (LLM)-guided policy evolution. PINNMorph maintains a population of state-conditioned adaptation policies that map execution diagnostics to controlled interventions over topology modification, additive representation augmentation, objective balancing, gradient handling, adaptive sampling, and optimizer-phase control. At each intervention opportunity, candidate programs are instantiated from the current policy population, selected according to the observed training state, and applied directly to the PINN under training. The resulting model inherits its existing parameters and training state and continues optimization along the same trajectory. Execution outcomes are subsequently used to evaluate interventions and evolve the policy population. Unlike pre-training architecture search or fixed adaptation rules, PINNMorph jointly adapts the current PINN and the policies governing its interventions using feedback from actual training. Experiments on 13 PDE benchmarks show that PINNMorph achieves lower solution errors than SA-PINN, ConFIG, RoPINN, HARMONIC, and PINNsAgent across all evaluated problems. Ablation studies further examine the effects of online adaptation, state-conditioned intervention selection, and execution-feedback-driven policy evolution.

Figures & tables

Figure 1: Overview of PINNMorph. At each adaptation point, PINNMorph diagnoses the current training state, uses a policy population to propose candidate actions, selects an executable action based on validity and state-conditioned scoring, and applies it online to the PINN under training. The resulting execution feedback is audited against the estimated no-intervention trajectory and used to evolve the policy population for subsequent adaptations.

PDE	SA-PINN	ConFIG	RoPINN	HARMONIC	PINNsAgent	PINNMorph
KS 1D	6.40E+0 $\pm$ 4.63E-2	1.47E+1 $\pm$ 4.60E-1	6.37E+0 $\pm$ 5.01E-1	5.55E+1 $\pm$ 5.88E+0	1.12E+0 $\pm$ 3.27E-2	1.05E+0 $\pm$ 3.44E-1
Burgers 1D	4.77E-4 $\pm$ 3.08E-5	2.47E-4 $\pm$ 1.27E-4	6.31E-4 $\pm$ 6.35E-5	1.32E-4 $\pm$ 2.13E-5	7.24E-5 $\pm$ 1.49E-5	6.48E-5 $\pm$ 3.29E-5
NS 2D CG	7.10E-3 $\pm$ 3.11E-4	2.46E-2 $\pm$ 7.81E-3	9.91E-3 $\pm$ 5.27E-3	5.34E-3 $\pm$ 4.84E-4	4.44E-3 $\pm$ 1.30E-3	1.47E-3 $\pm$ 4.13E-4
Heat 2D CG	3.41E-3 $\pm$ 6.07E-5	1.60E-2 $\pm$ 3.96E-3	4.92E-3 $\pm$ 8.93E-4	2.87E-2 $\pm$ 4.96E-4	3.61E-3 $\pm$ 8.71E-4	1.81E-3 $\pm$ 4.48E-4
Heat 2D MS	3.03E-4 $\pm$ 2.25E-5	3.18E-4 $\pm$ 2.99E-5	3.95E-4 $\pm$ 8.06E-6	4.20E-4 $\pm$ 6.47E-5	3.13E-4 $\pm$ 2.59E-5	2.34E-4 $\pm$ 2.21E-5
Heat 2D VC	1.45E-2 $\pm$ 5.17E-3	3.52E-2 $\pm$ 4.91E-3	4.70E-1 $\pm$ 4.86E-3	9.95E-2 $\pm$ 3.86E-2	5.62E-3 $\pm$ 3.33E-3	1.82E-3 $\pm$ 2.54E-4

Table 1: Solution MSE on

13

PDE benchmarks. Results are reported as mean

\pm

standard deviation over five independent runs. Lower is better.

Variant	Online Adaptation	State-Conditioned Selection	Policy Evolution	Feedback Evolution
No Morph	$\times$	$\times$	$\times$	$\times$
Random Selection	✓	$\times$	✓	✓
Random Evolution	✓	✓	✓	$\times$
PINNMorph	✓	✓	✓	✓

Table 2: Component configurations used in the ablation study.

PDE	No Morph	Random Selection	Random Evolution	PINNMorph
KS 1D	5.21E+0 $\pm$ 1.76E+0	6.24E+0 $\pm$ 1.87E+0	5.03E+0 $\pm$ 5.74E-1	1.05E+0 $\pm$ 3.44E-1
Burgers 1D	6.82E-4 $\pm$ 5.05E-4	3.62E-4 $\pm$ 1.35E-4	7.92E-4 $\pm$ 4.96E-4	6.48E-5 $\pm$ 3.29E-5
NS 2D CG	8.72E-3 $\pm$ 3.38E-3	9.46E-2 $\pm$ 1.10E-2	4.39E-3 $\pm$ 1.86E-3	1.47E-3 $\pm$ 4.13E-4
Heat 2D CG	5.17E-3 $\pm$ 6.43E-4	1.95E-2 $\pm$ 4.83E-3	4.01E-2 $\pm$ 5.46E-3	1.81E-3 $\pm$ 4.48E-4
Heat 2D MS	4.26E-3 $\pm$ 3.91E-4	3.89E-3 $\pm$ 2.72E-4	4.43E-3 $\pm$ 8.00E-5	2.34E-4 $\pm$ 2.21E-5
Heat 2D VC	3.43E-2 $\pm$ 1.67E-2	4.20E-2 $\pm$ 1.46E-2	4.94E-3 $\pm$ 2.07E-3	1.82E-3 $\pm$ 2.54E-4

Table 3: Component ablation results. Results are reported as MSE (mean

\pm

standard deviation) over five independent runs. Lower is better.

Figure 2: Representative training trajectories of PINNMorph. Training loss and the Decision and Audit physics scores are tracked throughout optimization, with the two physics scores computed on the fixed decision and audit sets, respectively.

Training state	Diagnostic signal	Action	Before $\rightarrow$ After
Localized error	Shock-localized residual	Localized Expert	$1.01\times 10^{-2}\rightarrow 6.32\times 10^{-4}$
Representation limit	Non-local residual	Fourier Branch	$3.71\times 10^{-3}\rightarrow 2.59\times 10^{-4}$
Spatial imbalance	Residual imbalance	Adaptive Sampling	$3.08\times 10^{-2}\rightarrow 7.63\times 10^{-3}$
Gradient conflict	Inter-equation conflict	Topology Split	$1.36\times 10^{-3}\rightarrow 6.53\times 10^{-4}$
Spatial imbalance	High-D residual imbalance	Adaptive Sampling	$2.26\times 10^{-3}\rightarrow 8.08\times 10^{-4}$
Optimization plateau	Stable gradients	Correction Branch	$4.86\times 10^{-3}\rightarrow 1.46\times 10^{-3}$

Table 4: Representative online adaptations of PINNMorph and their effects under different training states.

Variant	LLM Calls	LLM Tokens	Wall Time (s)	Median MSE Ratio
No Morph	0	0	1250	12.65 $\times$
Random Selection	30	$6.52\times 10^{4}$	1273	10.77 $\times$
Random Evolution	33	$1.06\times 10^{5}$	1286	10.30 $\times$
PINNMorph	38	$4.40\times 10^{5}$	1308	1.00 $\times$

Table 5: Computational cost and median MSE ratios of PINNMorph and its ablated variants. All variants use the same

20{,}000

-step optimization budget. Median MSE ratios are computed relative to PINNMorph.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Group	Mechanism	Target and execution rule	Parameters
Topology	Widen	Replicate units in selected hidden layers and divide outgoing weights by replication count (Net2Wider map).	Width fraction; layers.
Topology	Regrow	Expand selected hidden layers through the same function-preserving duplication map.	Width fraction; layers.
Topology	Add layer	Insert a new hidden layer after a selected layer; initialize its weights and continue the live model.	Layer index.
Topology	Prune	Remove low-utility hidden units while respecting a minimum width.	Width fraction; minimum width; layers.
Topology	Redistribute	Remove donor units and add capacity elsewhere in the hidden stack.	Move fraction; minimum width; layers.
Topology	Local reinitialization	Reinitialize a bounded fraction of units in selected hidden layers.	Reset fraction; layers.

Appendix

Table 6: Executable online mechanisms. “Zero output” means that the added module leaves the current function unchanged at insertion. Numeric arguments are optional; the executor derives defaults from the validated strength and diagnostic state.

Neural PDE Solvers

Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.

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Open paper AskPDF arXiv

Physics-informed neural networks (PINNs) require coordinated choices across representation, architecture, sampling, constraints, loss construction, and optimization, yet effective configurations can vary substantially across partial differential equations (PDEs). Existing automated design methods search over these choices, but training outcomes are still used primarily to rank candidates rather than to inform subsequent designs. We develop PINNForge, an execution-grounded large language model (LLM)-driven evolutionary framework that treats observed PINN training behavior as a cross-generation design signal. PINNForge initializes diverse configurations from PDE-related prior knowledge, executes them, retains globally strong configurations, and feeds their execution evidence and accumulated run-level experience back to the LLM. The LLM then revises and recombines coupled design components or explores new combinations within a generate--execute--evaluate--evolve loop. Across 25 PDE benchmarks, PINNForge achieves the lowest mean MSE on 24 tasks among RoPINN, PINNsFormer, PINNsAgent, and PINNForge. Removing knowledge guidance, execution feedback, or evolutionary search increases the median task-wise MSE ratio to 3.74

\times

, 12.10

\times

, and 10.10

\times

, respectively, relative to full PINNForge.

Abstract

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