Efficient Solution of Allen-Cahn Equation Considering Phase Boundary Thickness Variation Driven by Physics-Informed Neural Networks

作者：Xinyu Li, Menghan Wang, Yuxin Si, Zuoquan Zhu · 发表于：Science and Engineering · 年份：2026 · DOI：10.57237/j.se.2026.01.002

This study focuses on solving the Allen-Cahn (AC) equation with consideration of varying phase boundary thickness using Physics-Informed Neural Networks (PINN). Traditional numerical methods often exhibit declining computational efficiency and accuracy as problem complexity increases when handling complex phase boundary conditions; PINNs offer a novel approach to addressing such challenges. The study first conducts a detailed derivation of the AC equation and constructs a PINN-based solution model, including three key steps: designing a specific neural network architecture; building a loss function composed of partial differential equation (PDE) residual terms, boundary condition residual terms, and initial condition residual terms; and performing model training. The computational results demonstrate that the established PINN model can effectively solve the AC equation with varying phase boundary thicknesses, accurately revealing the distribution patterns of the order parameter under different phase boundary thicknesses, and adapt to the challenges brought by changes in phase boundary thickness by adjusting the number of iterations to achieve stable and efficient numerical solutions. The phase boundary thickness parameter has a significant impact on order parameter distribution, computational stability, and convergence. Additionally, a discussion is conducted between PINN and the finite difference method. This research provides support for in-depth understanding of the physic...