  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.
