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

TL;DR

This paper introduces a hybrid finite element/neural network solver for the Poisson problem, combining coarse grid solutions with neural network-predicted fine scale corrections, supported by theoretical analysis and numerical experiments.

Contribution

It formalizes a hybrid method integrating neural networks with finite element solutions and provides an a-priori error analysis and insights on training set size effects.

Findings

The hybrid method improves solution accuracy over traditional finite element methods.

The a-priori error analysis offers theoretical guarantees for the method.

Training set size significantly impacts the neural network's correction quality.

Abstract

We 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), (Margenberg et al., J Comput Phys 460:110983, 2022; A neural network multigrid solver for the Navier-Stokes equations) 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.