Physical informed memory networks for solving PDEs: implementation and applications

Abstract

Abstract With the advent of physics informed neural networks (PINNs), deep learning has gained interest for solving nonlinear partial differential equations (PDEs) in recent years. In this paper, physics informed memory networks (PIMNs) are proposed as a new approach to solving PDEs by using physical laws and dynamic behavior of PDEs. Unlike the fully connected structure of the PINNs, the PIMNs construct the long-term dependence of the dynamics behavior with the help of the long short-term memory network. Meanwhile, the PDEs residuals are approximated using difference schemes in the form of convolution filter, which avoids information loss at the neighborhood of the sampling points. Finally, the performance of the PIMNs is assessed by solving the KdV equation and the nonlinear Schrödinger equation, and the effects of difference schemes, boundary conditions, network structure and mesh size on the solutions are discussed. Experiments show that the PIMNs are insensitive to boundary conditions and have excellent solution accuracy even with only the initial conditions.