A deep First-Order System Least Squares method for solving elliptic PDEs. (1st January 2023)

• Record Type: Journal Article
• Authors: Bersetche, Francisco M.; Borthagaray, Juan Pablo
• Is Part Of: Computers & mathematics with applications. Volume 129(2023)
• Page Start: 136
• Page End: 150
• Publication Date: 2023-01-01
• Subjects: Elliptic PDEs -- Machine learning -- Deep learning -- High-dimensional PDEs; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2022.11.014
• Languages: English
• ISSNs: 0898-1221

Abstract

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 Γ-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.