关键词: 偏微分方程, 数值分析, 神经网络, 泰勒公式, 无网格

Abstract: Solving partial differential equations plays a vital role of numerical analysis in scientific and engineering fields such as computational fluid dynamics. Due to the multi-scale nature of physics and sensitivity to the quality of the discrete mesh, traditional numerical methods often require complex human-computer interaction and expensive meshing overhead, which limit their application to many real-time simulation and optimal design problems. This paper proposes an improved neural network-based method for solving partial differential equations, named TaylorPINN. It utilizes the universal approximation theorem of neural networks and the function-fitting capability of Taylor formula, and provides a mesh-free numerical solving process. Numerical experimental results on Helmholtz, Klein-Gordon, and Navier-Stokes equations demonstrate that TaylorPINN is able to approximate the underlying mapping relations between the coordinate inputs and quantities of interest, yielding an accurate prediction result. Compared with the widely used physics-informed neural network method, TaylorPINN improves the prediction accuracy by a factor of 3~20x across different numerical problems.

Key words: partial differential equation, numerical analysis, deep neural network, Taylor formula, mesh-free