A Solution Method for Differential Equations Based on Taylor PINN

Abstract

Based on deep neural network, elliptic partial differential equations in complex regions are solved. Accurate and effective strategies and numerical methods for elliptic partial differential equations are proposed by implementing deep feedforward artificial neural network, appropriate loss function solving strategy are constructed. The solution of an elliptic partial differential equation is obtained by iteratively learning the parameters of a neural network. Constructing a composite multi-layer radial basis function neural network can improve the real function approximation performance and operational accuracy of the constructed multi-layer radial basis function neural network. Use this high-precision composite multi-layer radial basis function neural network to solve partial differential equations. By providing specific examples of solving partial differential equations, the effectiveness of this method is tested. An improved partial differential equation solving method based on deep neural networks (Taylor PINN) has been proposed. This method utilizes the universal approximation theorem of deep neural networks and the function fitting ability of Taylor’s formula to achieve a meshless numerical solution process. The numerical experimental results on Helmholtz, Klein Gordon, and Navier Stokes equations show that Taylor PINN can well fit the mapping relationship between the coordinates of spatiotemporal points in the computational domain and the value of the desired function, which can provide accurate numerical prediction results. Compared with commonly used physical information based neural network methods, Taylor PINN improves prediction accuracy by 3–20 times for different numerical problems.

Keywords

• Partial differential equations
• numerical analysis
• neural network
• Taylor's formula
• loss function