A hybrid finite element/neural network solver and its application to the Poisson problem

Abstract

AbstractWe analyze a hybrid method that enriches coarse grid finite element solutions with fine scale fluctuations obtained from a neural network. The idea stems from the Deep Neural Network Multigrid Solver (DNN‐MG) which embeds a neural network into a multigrid hierarchy by solving coarse grid levels directly and predicting the corrections on fine grid levels locally (e.g., on small patches that consist of several cells) by a neural network. Such local designs are quite appealing, as they allow a very good generalizability. In this work, we formalize the method and describe main components of the a‐priori error analysis. Moreover, we numerically investigate how the size of training set affects the solution quality.