Quasi-optimal $hp$-finite element refinements towards singularities via deep neural network prediction

We show how to construct the deep neural network (DNN) expert to predict quasi-optimal $hp$-refinements for a given computational problem. The main idea is to train the DNN expert during executing the self-adaptive $hp$-finite element method ($hp$-FEM) algorithm and use it later to predict further $hp$ refinements. For the training, we use a two-grid paradigm self-adaptive $hp$-FEM algorithm. It employs the fine mesh to provide the optimal $hp$ refinements for coarse mesh elements. We aim to construct the DNN expert to identify quasi-optimal $hp$ refinements of the coarse mesh elements. During the training phase, we use the direct solver to obtain the solution for the fine mesh to guide the optimal refinements over the coarse mesh element. After training, we turn off the self-adaptive $hp$-FEM algorithm and continue with quasi-optimal refinements as proposed by the DNN expert trained. We test our method on three-dimensional Fichera and two-dimensional L-shaped domain problems. We verify the convergence of the numerical accuracy with respect to the mesh size. We show that the exponential convergence delivered by the self-adaptive $hp$-FEM can be preserved if we continue refinements with a properly trained DNN expert. Thus, in this paper, we show that from the self-adaptive $hp$-FEM it is possible to train the DNN expert the location of the singularities, and continue with the selection of the quasi-optimal $hp$ refinements, preserving the exponential convergence of the method.