r-Adaptive deep learning method for solving partial differential equations

We introduce a Deep Neural Network (DNN) method for solving Partial Differential Equations (PDEs) that simultaneously: (a) constructs an optimal r-adapted mesh, i.e., given an initial mesh, it provides optimal node locations, and (b) solves the PDE over the constructed r-adaptive mesh. The node locations are optimized over a set of 1D boundary nodes, and the corresponding 2D quadrilateral meshes are built using tensor product. The method supports the definition of fixed interfaces to create conforming meshes and allows nodes to jump across them, permitting topological variations. To numerically illustrate the performance of our novel r-adaptive deep learning method, we apply it in combination with other numerical methods including collocation, Least Squares, and the Deep Ritz method. We solve one- and two-dimensional problems whose solutions are smooth, singular, and/or exhibit strong gradients. Results consistently show the outperformance of employing r-adaptivity, while in some cases the improvement is limited by the tensor-product structure of the mesh.