$r-$Adaptive Deep Learning Method for Solving Partial Differential Equations

Autor:	Omella, Ángel J., Pardo, David
Rok vydání:	2022
Předmět:	Mathematics - Numerical Analysis Computer Science - Machine Learning Computer Science - Neural and Evolutionary Computing 65N50 68T07
Druh dokumentu:	Working Paper
Popis:	We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we build two- or three-dimensional meshes. The method allows the definition of fixed interfaces to design conforming meshes, and enables changes in the topology, i.e., some nodes can jump across fixed interfaces. The method simultaneously optimizes the node locations and the PDE solution values over the resulting mesh. To numerically illustrate the performance of our proposed $r-$adaptive method, we apply it in combination with a collocation method, a Least Squares Method, and a Deep Ritz Method. We focus on the latter to solve one- and two-dimensional problems whose solutions are smooth, singular, and/or exhibit strong gradients. Comment: 19 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2210.10900 Zobrazit plný text záznamu View this record from Arxiv