A deep first-order system least squares method for solving elliptic PDEs

TL;DR

This paper introduces a deep-learning-based First-Order System Least Squares method for solving second-order elliptic PDEs, capable of handling high-dimensional, variational, and non-variational problems with proven convergence.

Contribution

It develops a novel meshless deep-learning approach for elliptic PDEs with convergence proofs and broad applicability to various problem types.

Findings

Demonstrates effective numerical solutions for elliptic PDEs

Proves $\Gamma$-convergence of neural network approximations

Shows robustness in high-dimensional problem settings

Abstract

We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The method we propose is capable of dealing with either variational and non-variational problems, and because of its meshless nature, it can also deal with problems posed in high-dimensional domains. We prove the $\Gamma$-convergence of the neural network approximation towards the solution of the continuous problem, and extend the convergence proof to some well-known related methods. Finally, we present several numerical examples illustrating the performance of our discretization.