Physical informed memory networks based on domain decomposition for solving nonlinear partial differential equations

Abstract

In recent years, deep learning models have emerged as a popular numerical method for solving nonlinear partial differential equations (PDEs). In this paper, the improved physical informed memory networks (PIMNs) are introduced, which are constructed upon domain decomposition. In the improved PIMNs, the solution domain is decomposed into non-overlapping rectangular sub-domains. The loss for each sub-domain is computed independently, and an adaptive function is employed to dynamically adjust the coefficients of the loss terms. This approach significantly improves the PIMNs’ ability to train regions with high loss values. To validate the superiority of the improved PIMNs, the nonlinear Schrödinger equation, the KdV-Burgers equation, and the KdV-Burgers-Kuramoto equation are solved via both the original and the improved PIMNs. The experimental results clearly show that the improved PIMNs provide a significant enhancement in terms of solution accuracy compared to the original PIMNs.