Deep FDM: Enhanced finite difference methods by deep learning

Abstract

In this work, we propose a new idea to improve numerical methods for solving partial differential equations (PDEs) through a deep learning approach. The idea is based on an approximation of the local truncation error of the numerical method used to approximate the spatial derivatives of a given PDE. We present our idea as a proof of concept to improve the standard and compact finite difference methods (FDMs), but it can be easily generalized to other numerical methods.Without losing the consistency and convergence of the FDM numerical scheme, we achieve a higher numerical accuracy in the presented one- and two-dimensional examples, even for parameter ranges outside the trained region. We also perform a time complexity analysis and show the efficiency of our method.