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

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. 本研究聚焦于物理信息神经网络（PINN）求解不同相界厚度变化的Allen-Cahn（AC）方程。传统数值方法在处理复杂相边界条件时，其计算效率与精确度常随问题复杂度而下降；PINN为解决此类问题提供了新途径。本研究首先详细推导了AC方程，进而构建了基于PINN的求解模型，包括设计特定的神经网络架构、构建由偏微分方程残差项、边界条件残差项和初始条件残差项所组成的损失函数，并进行模型训练。计算结果表明，所建立的PINN模型能有效求解不同相界厚度变化的AC方程，准确揭示序参数在不同相界厚度下的分布规律，且能通过改变迭代次数适应相界厚度变化带来的挑战，实现稳定高效的数值求解。此外，将PINN计算结果与有限差分法进行了对比。本研究为深入理解AC方程物理内涵、优化数值计算方法提供支持，也为PINN在材料科学、凝聚态物理等领域解决类似复杂问题奠定基础。未来可进一步拓展PINN在高维相场模型、多物理场耦合问题中的应用，结合其他技术提高相场模拟精确度和效率，并开展更多实际应用案例研究。