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

• URL: http://arxiv.org/abs/2210.10900v1
• Date: Wed, 19 Oct 2022 21:38:46 GMT
• Title: $r-$Adaptive Deep Learning Method for Solving Partial Differential Equations
• Authors: \'Angel J. Omella and David Pardo
• Abstract summary: 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 optimize the boundary node locations in one dimension, from which we build two- or three-dimensional meshes.
• Score: 0.685316573653194
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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