[Paper Review] Physics-informed neural operator for predictive parametric phase-field modelling

Nanxi Chen, Airong Chen+1 more|arXiv (Cornell University)|Mar 10, 2026

Machine Learning in Materials Science0 citations

TL;DR

PF-PINO combines physics-informed constraints with Fourier neural operators to learn parametric phase-field models, producing more accurate and stable long-term predictions than data-driven FNO across several benchmarks.

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.

Motivation & Objective

• Motivate high computational cost of traditional phase-field simulations and the need for fast, accurate surrogates for parametric studies.
• Propose a physics-informed neural operator framework that embeds PDE residuals into training to enforce physical laws.
• Demonstrate improved accuracy, generalisation, and long-term stability over data-driven baselines across diverse phase-field problems.
• Showcase autoregressive rollout capabilities with physical constraints for stable long-duration predictions.

Proposed method

• Build on the Fourier neural operator (FNO) backbone to learn parametric phase-field solution operators.
• Embed PDE residuals of governing phase-field equations into the training loss as physics-informed constraints.
• Use an autoregressive scheme to map current state and parameter fields to the next time step, enabling trajectory generation.
• Compute PDE residuals via finite-difference or spectral differentiation and balance data fidelity with physics residuals in a composite loss.
• Optionally apply test-time physics-informed fine-tuning to minimise accumulated PDE violations over the trajectory.

Experimental results

Research questions

• RQ1 Can PF-PINO generalise across diverse parametric phase-field problems (electrochemical corrosion, dendritic solidification, spinodal decomposition) better than data-driven FNO?
• RQ2 Does embedding PDE residuals improve long-term autoregressive stability and interface accuracy under extrapolation and sparse data scenarios?
• RQ3 How do physical constraints affect generalisation to non-periodic boundaries and sharp interface dynamics in phase-field models?
• RQ4 What is the impact of physics-informed training on the capability to capture global spectral features and morphologies across time?

Key findings

• PF-PINO significantly outperforms FNO in relative L2 error across benchmarks, including 0.53% vs 1.58% for pencil-electrode corrosion under unseen parameters.
• PF-PINO reduces relative Hausdorff distance notably (e.g., 0.33 vs 0.83 in pencil-electrode corrosion) and demonstrates superior boundary and interface accuracy.
• In dendritic solidification, PF-PINO maintains accurate morphologies and outperforms FNO, including improved extrapolation to higher latent-heat coefficients.
• In spinodal decomposition, PF-PINO yields better structure-factor fidelity across the spectrum, particularly in low-k modes, and displays more stable autoregressive errors.
• Across benchmarks, PF-PINO achieves faster convergence during training and sustains lower error growth during long autoregressive rollouts.
• The study argues that physics-informed constraints regularise the solution space, enhancing extrapolation and reducing unphysical artefacts.