Efficient Solution of Allen-Cahn Equation Considering Phase Boundary Thickness Variation Driven by Physics-Informed Neural Networks
School of Mathematics and Statistics, Zhengzhou Normal University, Zhengzhou 450044, China
| Abstract: | 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 physical connotation of the AC equation and optimization of numerical calculation methods, and also lays a foundation for the application of PINN in solving similar complex problems in fields such as materials science and condensed matter physics. Future work can further expand the application of PINN in high-dimensional phase-field models and multi-physics coupling problems, combine other technologies to improve the accuracy and efficiency of phase-field simulations, and carry out more studies on practical application cases. |
| Keywords: | Physics-Informed Neural Networks; Allen-Cahn Equation; Phase Boundary Thickness; Order Parameter Distribution; Finite Difference Method |
| DOI: | 10.57237/j.se.2026.01.002 |
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Received: 14 October 2025
Accepted: 26 January 2026
Published Online: 6 February 2026
Funding
| 1. | 河南省自然科学基金项目 (252300420935) |
| 2. | 郑州师范学院大学生创新创业训练计划目 (DCY2024011) |
| 3. | 郑州师范学院青年骨干教师资助培养计划 (2026GGJS02) |
| 4. | 郑州师范学院科研启动专项经费 (2021-702442) |
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