Deep Ritz-Finite Element methods: Neural Network Methods trained with Finite Elements

TL;DR

This paper introduces a hybrid approach combining finite element methods with neural networks for solving elliptic PDEs on low-dimensional domains, improving efficiency and accuracy.

Contribution

The paper proposes a novel method that integrates finite element computations into neural network training for elliptic PDEs, enhancing stability and convergence.

Findings

The method is stable and converges to PDE solutions.

Numerical results show improved efficiency and robustness.

The approach leverages finite element tools within neural network training.

Abstract

While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains $\Omega \subset \mathbb{R} ^d, $ $d=1,2,3$ in association with more standard finite elements. We suggest to connect finite elements and neural network approximations through training, i.e., using finite element spaces to compute the integrals appearing in the loss functionals. This approach, retains the simplicity of classical neural network methods for PDEs, uses well established finite element tools (and software) to compute the integrals involved and it gains in efficiency and accuracy. We demonstrate that the proposed methods are stable and furthermore, we establish that the resulting approximations converge to the solutions of the PDE. Numerical results indicating the efficiency and robustness of the proposed algorithms are presented.