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

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.

arXiv comment15 pages, 6 figures, 5 tables, accepted by Journal of Computational Physics

References in corpus (5)

• Adam: A Method for Stochastic Optimization
• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• DGM: A deep learning algorithm for solving partial differential equations
• A unified deep artificial neural network approach to partial differential equations in complex geometries
• A Discussion on Solving Partial Differential Equations using Neural Networks

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