Can Neural Networks learn Finite Elements?

TL;DR

This paper investigates whether neural networks can replicate finite element solutions for simple boundary value problems, aiming to understand their potential and limitations in approximating PDEs.

Contribution

It constructs a neural network that reproduces finite element solutions as a minimum of its cost function, providing insights into neural network capabilities for PDE approximation.

Findings

Neural networks can be designed to reproduce finite element solutions.

The study highlights challenges in using neural networks for PDE approximation.

Insights into the limitations of neural networks in this context.

Abstract

The aim of this note is to construct a neural network for which the linear finite element approximation of a simple one dimensional boundary value problem is a minimum of the cost function to find out if the neural network is able to reproduce the finite element approximation. The deepest goal is to shed some light on the problems one encounters when trying to use neural networks to approximate partial differential equations