Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs

TL;DR

This paper introduces an unsupervised deep learning approach using least-squares functionals to numerically solve elliptic PDEs, demonstrating effectiveness on one-dimensional second-order elliptic problems.

Contribution

It develops a novel deep neural network method based on first-order system least-squares functionals for solving elliptic PDEs without supervision.

Findings

Effective approximation of elliptic PDE solutions in 1D

Utilizes least-squares functionals as loss functions

Shows promising numerical results

Abstract

This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine parameters of the deep neural network. There are various least-squares functionals for a partial differential equation. This paper focuses on the so-called first-order system least-squares (FOSLS) functional studied in [3], which is based on a first-order system of scalar second-order elliptic PDEs. Numerical results for second-order elliptic PDEs in one dimension are presented.