基于物理信息神经网络的Allen-Cahn方程应用研究

Study on the Application of the Allen-Cahn Equation Based on Physics-Informed Neural Networks

摘要: 近年来，随着深度学习这一领域的不断崛起与发展，其中的物理信息神经网络(PINN)可以通过其无网格化的求解框架实现对于偏微分方程的精确求解功能。基于上述观点，本文提出以Allen-Cahn方程为研究模型，结合PINN中的激活函数和训练次数等要素对方程进行分析，研究在不同参数设置下解的误差情况，并得出相对最优的配置方案。此外，本文还针对其中的扩散系数D进行反演，发现随着训练次数的增加，反演的参数在不断逼近真实值。此外，PINN在反演时不依赖初始值的设定，无论距离真实值距离远或近都可以实现高精度的反演。

Abstract: In recent years, with the rapid rise and development of deep learning, physics-informed neural networks (PINNs) have emerged as a powerful tool for solving partial differential equations through their mesh-free framework. Based on this perspective, this paper takes the Allen-Cahn equation as a model problem and analyzes the solution accuracy by incorporating factors such as activation functions and training iterations within PINNs. The study investigates the error behavior under various parameter settings and identifies relatively optimal configurations. Furthermore, the paper conducts inversion of the diffusion coefficient D and finds that as the number of training iterations increases, the inverted parameters progressively approach the true values. Notably, PINNs do not rely on initial guesses during inversion; they achieve high-precision results regardless of how far or close the initial values are to the actual ones.