Neural-Network-Assisted Finite Difference Discretization for Numerical Solution of Partial Differential Equations

Abstract

A neural-network-assisted numerical method is proposed for the solution of Laplace and Poisson problems. Finite differences are applied to approximate the spatial Laplacian operator on nonuniform grids. For this, a neural network is trained to compute the corresponding coefficients for general quadrilateral meshes. Depending on the position of a given grid point x0 and its neighbors, we face with a nonlinear optimization problem to obtain the finite difference coefficients in x0. This computing step is executed with an artificial neural network. In this way, for any geometric setup of the neighboring grid points, we immediately obtain the corresponding coefficients. The construction of an appropriate training data set is also discussed, which is based on the solution of overdetermined linear systems. The method was experimentally validated on a number of numerical tests. As expected, it delivers a fast and reliable algorithm for solving Poisson problems.

Keywords

• boundary value problems
• Laplacian
• Poisson problem
• neural networks
• finite difference discretization
• learning data