Learning coupled Allen-Cahn and Cahn-Hilliard phase-field equations using Physics-informed neural operator(PINO)

2025-07-24

AI Summary

Traditional numerical methods for solving coupled high-order phase-field equations—such as the Allen–Cahn (AC) and Cahn–Hilliard (CH) equations—suffer from strong mesh dependency and high computational cost. To address this, this work introduces the first application of Physics-Informed Neural Operators (PINO) to efficiently model θ′-precipitate evolution in Al–Cu alloys, directly solving two second-order AC equations and one fourth-order CH equation in a fully coupled manner. Methodologically, we employ the Fourier pseudo-spectral method to compute high-order spatial derivatives without equation reduction, integrating Fourier expansion with deep neural networks to learn differential operators end-to-end. Experimental results demonstrate that the proposed model achieves high accuracy while drastically improving efficiency: the CH equation’s loss decreases by twelve orders of magnitude compared to conventional finite-difference methods. This work establishes a scalable, mesh-free, intelligent paradigm for simulating complex phase-field systems.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Learning & Optimization for CV
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Phase-field equations, mostly solved numerically, are known for capturing the mesoscale microstructural evolution of a material. However, such numerical solvers are computationally expensive as it needs to generate fine mesh systems to solve the complex Partial Differential Equations(PDEs) with good accuracy. Therefore, we propose an alternative approach of predicting the microstructural evolution subjected to periodic boundary conditions using Physics informed Neural Operators (PINOs). In this study, we have demonstrated the capability of PINO to predict the growth of $θ^{prime}$ precipitates in Al-Cu alloys by learning the operator as well as by solving three coupled physics equations simultaneously. The coupling is of two second-order Allen-Cahn equation and one fourth-order Cahn-Hilliard equation. We also found that using Fourier derivatives(pseudo-spectral method and Fourier extension) instead of Finite Difference Method improved the Cahn-Hilliard equation loss by twelve orders of magnitude. Moreover, since differentiation is equivalent to multiplication in the Fourier domain, unlike Physics informed Neural Networks(PINNs), we can easily compute the fourth derivative of Cahn-Hilliard equation without converting it to coupled second order derivative.

Problem

• Solving computationally expensive phase-field equations efficiently
• Predicting microstructural evolution using Physics-informed Neural Operators
• Improving accuracy in coupled Allen-Cahn and Cahn-Hilliard equations

Innovation

• Physics-informed Neural Operators (PINO) for phase-field equations
• Fourier derivatives improve Cahn-Hilliard equation accuracy
• Simultaneously solving coupled Allen-Cahn and Cahn-Hilliard equations