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

TL;DR

This paper presents an $r$-adaptive deep learning approach for solving PDEs that optimizes mesh node locations and solution values simultaneously, improving accuracy for complex solution behaviors.

Contribution

It introduces an $r$-adaptive algorithm that optimizes mesh node positions and solution values in deep neural network-based PDE solvers, allowing for conforming and topology-changing meshes.

Findings

Effective in handling smooth and singular solutions.

Compatible with collocation, Least Squares, and Deep Ritz methods.

Demonstrates improved accuracy with adaptive meshes.

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.