Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution

Nanxi Chen; Airong Chen; Rujin Ma

cs.LGMar 10, 2026

Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution

Authors: Nanxi Chen, Airong Chen, Rujin Ma

Organizations: College of Civil Engineering, Tongji University, Shanghai 200092, China

Abstract

Predicting interfacial degradation and microstructural evolution through phase-field modelling is computationally intensive, particularly for parametric studies. While neural operators such as the Fourier neural operator (FNO) offer a promising route to acceleration, the absence of physical constraints compromises their generalisation and long-term stability. Here, we present PF-PINO, a physics-informed neural operator framework that embeds phase-field governing equation residuals directly into the training loss, together with gradient normalisation loss balancing and a staggered training scheme for coupled multi-field systems. PF-PINO is validated on four benchmarks including pencil-electrode corrosion, electro-polishing corrosion, dendritic solidification, and spinodal decomposition, collectively probing parametric generalisation across orders-of-magnitude kinetic regimes, non-periodic boundaries with stochastic morphologies, coupled multi-field dynamics with out-of-distribution extrapolation, and long-term autoregressive stability under complex pattern formation. Across all benchmarks, PF-PINO reduces relative $L^2$ errors by up to an order of magnitude over FNO and achieves sub-mesh-size interface accuracy, establishing a data-efficient surrogate for high-throughput parametric phase-field assessment in computational materials science and engineering applications.

Figures & tables

Figure 1: Architecture and training strategy of the physics-informed neural operator framework for phase-field problems . Top : The PF-PINO framework employs an autoregressive scheme to model temporal evolution of phase-field systems. The Fourier neural operator maps the current state $\bm{u}(\bm{x},t_{n})$ and static parameter fields $\bm{a}(\bm{x})$ to the subsequent state $\bm{u}(\bm{x},t_{n+1})$ through lifting projection ( $\mathcal{P}$ ), sequential spectral convolutions, and output projection ( $\mathcal{Q}$ ). During inference, recursive application generates complete solution trajectories from initial conditions $\bm{u}(\bm{x},t_{0})$ to time $T$ . Bottom left : Spectral convolution architecture. Input features undergo Fourier transformation ( $\mathcal{F}$ ), multiplication with learnable spectral filters ( $\mathcal{R}$ ) in frequency space, and inverse transformation ( $\mathcal{F}^{-1}$ ). This spectral branch is combined with a linear bypass branch ( $\mathcal{W}$ ) before nonlinear activation ( $\sigma$ ), enabling efficient learning of solution operators in both spectral and physical domains. Bottom right : Composite loss function for physics-informed training. One-step training minimises weighted combination of data fidelity loss $\mathcal{L}_{\mathrm{d}}$ (measuring prediction accuracy against reference solutions) and PDE residual loss $\mathcal{L}_{\mathrm{p}}$ (enforcing governing equation compliance). Gradient normalisation and a staggered training scheme are employed to stabilise optimisation and mitigate gradient conflicts in coupled multi-field systems. Optional rollout fine-tuning further reduces accumulated PDE violations across the full temporal trajectory $t_{0}\to t_{n}$ .

		FNO	PF-PINO	FNO	PF-PINO
Benchmark test	Parametrisation	Rel. $L^{2}$ error (in $\%$ )		Rel. Hausdorff distance
Pencil-electrode corrosion	Interface kinetics $L$	1.58	0.53	0.83	0.33
Electro-polishing corrosion	Initial interface profile	22.02	1.44	17.0	1.0
Dendritic crystal solidification	Latent heat coeff. $K$	4.85	1.72	4.81	1.10
Spinodal decomposition	Mobility $M$ & perturbation	19.10	9.71	1.98	1.56

Table 1: Summary of benchmark results obtained using FNO and PF-PINO models. The relative

L^{2}

errors are computed on the full spatio-temporal domain and averaged across all variable channels and test cases. The relative Hausdorff distance quantifies the accuracy of predicted interface locations for phase-field variables, computed at the final time step, normalised by the mesh size to provide a dimensionless measure of spatial accuracy, and averaged across all test cases.

Figure 2: Pencil-electrode corrosion simulation. a Problem schematic of the 1D pencil-electrode corrosion model with parametric interface kinetics coefficient $L$ . b Representative one-step predictions of phase-field variable $\phi$ (validation set) compared with FEM reference solutions. c Training convergence: relative $L^{2}$ error versus epoch for PF-PINO and FNO. d Autoregressive error accumulation: step-wise relative $L^{2}$ error evolution over the full prediction horizon. e Spatio-temporal phase-field distributions (reference, PF-PINO prediction, absolute error) for test cases with varying $L$ values. Spatial domain (horizontal axis) and temporal evolution (vertical axis) obtained through 100-step autoregressive rollouts.

Figure 3: Electro-polishing corrosion with parametric initial interface profiles. a Problem schematic: 2D corrosion model with sinusoidally parametrised initial metal-electrolyte interface. b Training convergence: relative $L^{2}$ error versus epoch. c Autoregressive stability: step-wise error evolution showing PF-PINO’s minimal accumulation versus FNO’s divergence. d Spatial distributions of phase-field variable $\phi$ at final time for a representative test case (reference, PF-PINO and FNO predictions, absolute errors). Results obtained through 100-step autoregressive predictions highlight the accuracy and reduced boundary artefacts of PF-PINO compared to FNO.

Figure 4: Dendritic crystal solidification with parametrised latent heat coefficient. a Training convergence: averaged relative $L^{2}$ error for phase-field variable $\phi$ and temperature $T$ . b Crystallised area fraction evolution for different latent heat coefficients $K$ , comparing PF-PINO and FNO predictions with FEM reference. c Autoregressive error accumulation: step-wise relative $L^{2}$ error showing PF-PINO’s stability versus FNO’s divergence. d Spatial distributions of $\phi$ and $T$ at final time ( $t=$10.0\text{\,}\mathrm{s}$$ ) for test cases (reference, predictions, absolute errors). Results for $K=0.9,\,1.3$ (interpolation) and $K=1.7,\,2.0$ (extrapolation) demonstrate PF-PINO’s superior accuracy in both regimes. — Figure 4: Dendritic crystal solidification with parametrised latent heat coefficient. a Training convergence: averaged relative $L^{2}$ error for phase-field variable $\phi$ and temperature $T$ . b Crystallised area fraction evolution for different latent heat coefficients $K$ , comparing PF-PINO and FNO predictions with FEM reference. c Autoregressive error accumulation: step-wise relative $L^{2}$ error showing PF-PINO’s stability versus FNO’s divergence. d Spatial distributions of $\phi$ and $T$ at final time ( $t=$ 10.0\text{,}\mathrm{s}$$ ) for test cases (reference, predictions, absolute errors). Results for $K=0.9,\,1.3$ (interpolation) and $K=1.7,\,2.0$ (extrapolation) demonstrate PF-PINO’s superior accuracy in both regimes.

Figure 5: Spinodal decomposition with parametrised mobility and initial perturbations. a Training convergence: relative

L^{2}

error versus epoch, illustrating data-driven training followed by physics-informed fine-tuning. b Autoregressive error evolution: step-wise relative

L^{2}

error demonstrating the superior stability of PF-PINO. c Radially averaged structure factor

S(k)

at the final time step averaged across all test cases. The close alignment of the PF-PINO profile with the reference in the low-

k

regime highlights its superior capture of global thermodynamic features. d Spatial distributions of

c

at sequential time steps for the test case with

M=1.2

(reference, predictions, absolute errors). The results show the accurate capture of phase separation and domain coarsening dynamics by PF-PINO.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A1: An example of the initial concentration field for the spinodal decomposition benchmark, generated by band-limited Fourier modes.

Notation	Description	Value
$L$	Interface kinetic coefficient	Parametrised
$\alpha_{\phi}$	Gradient energy coefficient	$1.02\text{\times}{10}^{-4}$
$w_{\phi}$	Height of the double well potential	$1.76\text{\times}{10}^{7}$
$\ell$	Interface thickness	$1.0\text{\times}{10}^{-5}$
$M$	Diffusivity parameter	$7.94\text{\times}{10}^{-18}$
$\mathcal{A}$	Free energy density-related parameter	$5.35\text{\times}{10}^{7}$

Appendix

Table A1: Physical parameters used in the phase-field corrosion modelling (SI units). In 1D pencil-electrode corrosion test, the interface kinetic coefficient

L

is parametrised over a wide range to capture different corrosion regimes, while in 2D electro-polishing corrosion test,

L

is fixed at

1\text{\times}{10}^{-10}\text{\,}{\mathrm{m}}^{3}

Notation	Description	Value
$K$	Latent heat coefficient	Parametrised
$\rho$	Mobility parameter	$1.0\text{\times}{10}^{3}$
$\varepsilon$	Interface width parameter	$0.015$
$\lambda$	Linear kinetic coefficient	$4.0\text{\times}{10}^{2}$
$D$	Temperature diffusivity	$2.5\text{\times}{10}^{-3}$
$\sigma$	Anisotropy strength parameter	$0.1$

Appendix

Table A2: Physical parameters used in the phase-field solidification modelling (Non-dimensional units).

Group	Hyperparameter	Corrosion1d	Corrosion2d	Solidification	Spinodal Decomposition
Network	In-channels	5	5	5	4
	Out-channels	2	2	2	1
	Depth	4	4	4	3
	Width	64	64	64	64
	Modes	8	(8,8)	(16,16)	(32,32)
	Activation	GeLU	ReLU	ReLU	GeLU

Appendix

Table A3: Hyperparameter configurations for all benchmark tests.

Figure A2: Pencil-electrode corrosion: One-step training and validation loss curves during PF-PINO and FNO training.

Figure A3: Pencil-electrode corrosion: One-step training loss curves of the Allen-Cahn and Cahn-Hilliard equations during PF-PINO training.

Figure A4: Pencil-electrode corrosion: Normalised boundary-work-corrected energy change during 100-step autoregressive prediction. Curves show the mean across the six unseen test cases, and shaded bands indicate one standard deviation.

Figure A5: Electro-polishing corrosion: One-step training and validation loss curves during PF-PINO and FNO training.

Figure A6: Electro-polishing corrosion: One-step training loss curves of the Allen-Cahn and Cahn-Hilliard equations during PF-PINO training.

Figure A7: Electro-polishing corrosion: Wall-clock costs of PF-PINO and FNO training, PF-PINO inference, and FEM simulation as the number of samples increases. PF-PINO inference and FEM simulation times correspond to generating complete trajectories at each sample count; PF-PINO uses 100 autoregressive updates with

\Delta t=

200\text{,}\mathrm{s}$$ per trajectory. Model training and inference use one NVIDIA GeForce RTX 5090 GPU, while FEM simulations use either a non-MPI process or

16

-core MPI parallelisation on a CPU node with an Intel Xeon Platinum 8470Q processor.

Figure A8: Electro-polishing corrosion: Time-step sensitivity of PF-PINO over the full autoregressive prediction horizon. The left panel shows the test relative

L^{2}

errors of PF-PINO and FEM for different time steps, evaluated against FEM trajectories generated with

\Delta t=

50\text{,}\mathrm{s}$$ . The right panel shows the step-wise test relative

L^{2}

errors for five representative time steps.

Figure A9: Electro-polishing corrosion: Phase-field variable $\phi$ snapshots of the predicted and ground truth solution fields at different time steps during 100-step autoregressive prediction with different initial interface morphologies. The prediction is obtained via the PF-PINO model.

Figure A10: Dendritic crystal solidification: One-step training and validation loss curves during PF-PINO and FNO training.

Figure A11: Dendritic crystal solidification: One-step training loss curves of the phase-field Allen-Cahn equation and heat equation during PF-PINO training.

Figure A12: Dendritic crystal solidification: Complete ablation of gradient normalisation and staggered training. Left: test relative

L^{2}

error during training. Right: step-wise relative

L^{2}

error during 200-step autoregressive prediction. The abbreviations indicate which components are included in each setting: PF-PINO uses both gradient normalisation and staggered training; W/o stagger uses gradient normalisation but not staggered training; W/o GradNorm uses staggered training but not gradient normalisation; Plain PINO uses neither gradient normalisation nor staggered training.

Setting	Gradient normalisation	Staggered training	Test rel. $L^{2}$ error (in %)
PF-PINO	✓	✓	$1.72$
W/o stagger	✓	✗	$2.55$
W/o GradNorm	✗	✓	$10.75$
Plain PINO	✗	✗	$14.02$

Appendix

Table A4: Dendritic crystal solidification: Final-epoch test relative

L^{2}

error for the complete ablation of gradient normalisation and staggered training. The abbreviations are the same as in Figure A12 .

Figure A13: Dendritic crystal solidification: Spatial distribution snapshots of the phase-field variable $\phi$ during solidification at different time steps with the latent heat coefficient $K=1.3$ .

Figure A14: Dendritic crystal solidification: Spatial distribution snapshots of the temperature field $T$ during solidification at different time steps with the latent heat coefficient $K=1.3$ .

Figure A15: Dendritic crystal solidification: Spatial distribution snapshots of the phase-field variable $\phi$ at the final time step ( $t=$10.0\text{\,}\mathrm{s}$$ ) predicted by the FNO model trained on a dataset with extended latent heat coefficient range $K\in\{0.8,1.0,1.4,1.6,2.2\}$ . — Figure A15: Dendritic crystal solidification: Spatial distribution snapshots of the phase-field variable $\phi$ at the final time step ( $t=$ 10.0\text{,}\mathrm{s}$$ ) predicted by the FNO model trained on a dataset with extended latent heat coefficient range $K\in\{0.8,1.0,1.4,1.6,2.2\}$ .

Figure A16: Spinodal decomposition: One-step training and validation loss curves during FNO training.

Figure A17: Spinodal decomposition: 100-step autoregressive fine-tuning loss curves during PF-PINO training.

Figure A18: Spinodal decomposition: Absolute concentration-mass drift during 100-step autoregressive prediction. The first plotted state corresponds to

t=\Delta t

. Curves show the mean across the five unseen test cases, and shaded bands indicate one standard deviation.

Figure A19: Spinodal decomposition: Spatial distribution of the concentration field $c$ at the final time step obtained via 100-step autoregressive prediction for different mobilities and initial concentration perturbations.

Physics-Informed Neural Operators

Physics-informed neural operators learn mappings between physical fields while embedding governing equations or structural constraints for faster, more reliable simulation.

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On the training of physics-informed neural operators for solving parametric partial differential equations

Nanxi Chen, Chuanjie Cui, Airong Chen, Sifan Wang+1

College of Civil Engineering, Tongji University, Shanghai 200092 · Department of Engineering Science, University of Oxford, Oxford OX1 3PJ · Institute for Foundations of Data Science, Yale University, New Haven, CT 06520

Neural Network Optimization Gradient Interference PDE Operator Learning Physics-Informed ML Physics-Informed Neural Operators

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Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross-instance generalization of neural operators with the data efficiency of physics-informed learning. Despite this promise, how to train PINOs efficiently and robustly remains less well-understood than the training of either data-driven neural operators or physics-informed neural networks (PINNs). To bridge this gap, we examine key components of the PINO training pipeline, including architecture design, optimizer choice, loss balancing, and collocation-point sampling strategy. We study three representative operator backbones, Deep Operator Network (DeepONet), Fourier Neural Operator (FNO), and Continuous Vision Transformer (CViT), across five diverse parametric PDE systems. Our results show that CViT provides consistently strong and stable performance across the considered benchmarks. Beyond architecture, we find that several optimization pathologies previously identified in PINN training naturally arise in PINOs, including gradient conflicts and causal violation. We also find that mitigation algorithms developed for PINNs remain effective in the PINO setting. We further compare physics-informed and data-driven training under different data regimes, revealing that a carefully designed physics-informed training pipeline can match, and in some cases, outperform purely data-driven neural operators. Taken together, these findings provide a systematic empirical understanding of the optimization challenges in PINO training and inform a practical pipeline for efficient and robust physics-informed operator learning. Code and data are available at https://github.com/NanxiiChen/PI-CViT.

Abstract

Physics-Informed Neural Operators

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