Numerical simulation of time fractional Allen-Cahn equation based on Hermite neural solver

Abstract

In this paper, we introduce a high-precision Hermite neural network solver which employs Hermite interpolation technique to construct high-order explicit approximation schemes for fractional derivatives. By automatically satisfying the initial conditions, the construction process of the objective function is simplified, thereby reducing the complexity of the solution. Our neural networks are trained and fine-tuned to solve one-dimensional (1D) and two-dimensional (2D) time fractional Allen-Cahn equations with limited sampling points, yielding high-precision results. Additionally, we tackle the parameter inversion problem by accurately recovering model parameters from observed data, thereby validating the efficacy of the proposed algorithm. We compare the $L^2$ relative error between the exact solution and the predicted solution to verify the robustness and accuracy of the algorithm. This analysis confirms the reliability of our method in capturing the fundamental dynamics of equations. Furthermore, we extend our analysis to three-dimensional (3D) cases, which is covered in the appendix, and provide a thorough evaluation of the performance of our method. This paper also conducts comprehensive analysis of the network structure. Numerical experiments indicate that the number of layers, the number of neurons in each layer, and the choice of learning rate play a crucial role in the performance of our solver.