Physics-informed neural operator for predictive parametric phase-field modelling

Abstract

Predicting the microstructural and morphological evolution of materials through phase-field modelling is computationally intensive, particularly for high-throughput parametric studies. While neural operators such as the Fourier neural operator (FNO) show promise in accelerating the solution of parametric partial differential equations (PDEs), the lack of explicit physical constraints, may limit generalisation and long-term accuracy for complex phase-field dynamics. Here, we develop a physics-informed neural operator framework to learn parametric phase-field PDEs, namely PF-PINO. By embedding the residuals of phase-field governing equations into the data-fidelity loss function, our framework effectively enforces physical constraints during training. We validate PF-PINO against benchmark phase-field problems, including electrochemical corrosion, dendritic crystal solidification, and spinodal decomposition. Our results demonstrate that PF-PINO significantly outperforms conventional FNO in accuracy, generalisation capability, and long-term stability. This work provides a robust and efficient computational tool for phase-field modelling and highlights the potential of physics-informed neural operators to advance scientific machine learning for complex interfacial evolution problems.

Figures (18)

• 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 linearly 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). Optional rollout fine-tuning further reduces accumulated PDE violations across the full temporal trajectory $t_0 \to t_n$.
• 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 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, 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 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 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 ($t=\qty{10.0}{s}$) 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 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.