Quasi-optimal

We show how to construct a deep neural network (DNN) expert to predict quasi-optimal hp -refinements for a given finite element problem in presence of singularities. The main idea is to train the DNN expert during the execution of 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. 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. We show that, from the self-adaptive hp -FEM, it is possible to train the DNN expert to predict the location of the singularities and continue with the selection of the quasi-optimal hp -refinements, preserving the exponential convergence of the method.