Error analysis for hybrid finite element/neural network discretizations

TL;DR

This paper introduces a hybrid finite element and neural network approach for accurately predicting fine-scale solutions of PDEs, with analysis of error, stability, and generalization capabilities.

Contribution

It presents a novel hybrid method combining finite elements and neural networks, including error and stability analysis, and demonstrates its effectiveness and generalization in numerical experiments.

Findings

Neural networks can locally correct coarse finite element solutions.

The method's accuracy depends on the size of the training set.

The approach generalizes to different test and training domains.

Abstract

We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The network is capable of locally correcting a coarse finite element solution towards a fine solution taking the source term and the coarse approximation as input. Key observation is the dependency between quality of predictions and the size of training set which consists of different source terms and corresponding fine & coarse solutions. We provide the a priori error analysis of the method together with the stability analysis of the neural network. The numerical experiments confirm the capability of the network predicting fine finite element solutions. We also illustrate the generalization of the method to problems where test and training domains differ from each other.