A deep learning-based FOSLS method for second-order elliptic PDEs
Speaker: Juan Pablo Borthagaray (Instituto de Matemática y Estadística, Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay)
Title: A deep learning-based FOSLS method for second-order elliptic PDEs.
Time: Wednesday, 2023.09.20, 2:00 p.m. (CET)
Place: fully virtual (contact Jakub Lengiewicz to register)
Format: 30 min. presentation + 30 min. discussion
Abstract: We present a First-Order System Least Squares (FOSLS) method based on deep learning for the numerical solution of second-order elliptic PDEs. The method we propose can handle both variational and non-variational problems. Due to its meshless nature, it is also suitable for tackling problems in high-dimensional domains. We prove the Γ-convergence of the neural network approximation towards the solution of the continuous problem. Furthermore, we extend the convergence proof to encompass several well-known related methods. Finally, we provide several numerical examples that illustrate the performance of the method.
Juan Pablo Borthagaray is an associate professor at Instituto de Matemática y Estadística, Facultad de Ingeniería, Universidad de la República in Uruguay. He obtained his PhD in Mathematics from the Universidad de Buenos Aires in 2017. His research area lies between the numerical analysis and the analysis of partial differential equations (PDEs). This includes mainly finite element methods, the analysis and design of numerical methods for nonlocal operators and the study of certain geometric PDEs.
Francisco M. Bersetche, Juan Pablo Borthagaray, A deep First-Order System Least Squares method for solving elliptic PDEs, Computers & Mathematics with Applications, vol. 129, pp. 136-150, 2023