A deep learning approach for solving Poisson’s equations

Abstract

Partial differential equations (PDEs) have a lot of applications in different fields of research during the last decades. In this paper, we study a mesh-free deep learning method for solving PDE systems, especially for Poisson’s equations. Different from traditional techniques using finite volume or finite element method, we design suitable neural networks that can approximate solutions of a PDE hy formulating it as an optimization problem. To minimize a loss function, we use the gradient descent algorithm to obtain the neural networks’ optimal set of parameters. The experimental results show that the proposed methods can achieve promising results in solving three types of PDEs: Burgers’ equation, Laplace’s equation, and Poisson’s equation, where the mean square errors vary from 10-7 to 10-10.