An Energy Minimization-Based Deep Learning Approach with Enhanced Stability for the Allen-Cahn Equation

The Allen-Cahn equation is a fundamental model in materials science for describing phase separation phenomena. This paper introduces an Energy-Stabilized Scaled Deep Neural Network (ES-ScaDNN) framework to solve the Allen-Cahn equation by energy minimization. Unlike traditional numerical methods, our approach directly approximates the solution of steady-state solution the Allen-Cahn equation by minimizing the associated energy functional using a deep neural network. ES-ScaDNN incorporates two key innovations. The first is a scaling layer designed to map the network output to the physical range of the Allen-Cahn phase variable. The second is a variance-based regularization term designed to promote clear phase separation. We demonstrate the accuracy and efficiency of ES-ScaDNN through comprehensive numerical experiments in both one and two dimensions. Our results show that ReLU activation functions are particularly well-suited for one-dimensional cases, while tanh functions are more suitable for two-dimensional problems due to their superior ability to maintain solution smoothness. Furthermore, we investigate how training epochs and the interface parameter ε influence the behavior of the solution. ES-ScaDNN provides a novel, accurate, and efficient deep learning framework for solving the Allen-Cahn equation, paving the way for tackling more complex phase-field problems.