DNN Approximation of Nonlinear Finite Element Equations

TL;DR

This paper explores using deep neural networks to approximate nonlinear operators in finite element methods, aiming to reduce computational costs in solving nonlinear PDEs by replacing coarse nonlinear operators with trained DNNs.

Contribution

The paper introduces a novel approach of employing DNNs to approximate nonlinear mappings in finite element discretizations, demonstrating its effectiveness in a two-level FAS scheme.

Findings

DNNs can effectively approximate nonlinear operators in finite element discretizations.

Replacing coarse nonlinear operators with DNNs reduces computational effort.

The approach is validated on a nonlinear diffusion-reaction PDE.

Abstract

We investigate the potential of applying (D)NN ((deep) neural networks) for approximating nonlinear mappings arising in the finite element discretization of nonlinear PDEs (partial differential equations). As an application, we apply the trained DNN to replace the coarse nonlinear operator thus avoiding the need to visit the fine level discretization in order to evaluate the actions of the true coarse nonlinear operator. The feasibility of the studied approach is demonstrated in a two-level FAS (full approximation scheme) used to solve a nonlinear diffusion-reaction PDE.