Refinement of polygonal grids using Convolutional Neural Networks with applications to polygonal Discontinuous Galerkin and Virtual Element methods

We propose new strategies to handle polygonal grids refinement based on Con-volutional Neural Networks (CNNs). We show that CNNs can be successfullyemployed to identify correctly the “shape” of a polygonal element so as to designsuitable refinement criteria to be possibly employed within adaptive refinementstrategies. We propose two refinement strategies that exploit the use of CNNsto classify elements’ shape, at a low computational cost. We test the proposedidea considering two families of finite element methods that support arbitrarilyshaped polygonal elements, namely Polygonal Discontinuous Galerkin (PolyDG)methods and Virtual Element Methods (VEMs). We demonstrate that theproposed algorithms can greatly improve the performance of the discretizationschemes both in terms of accuracy and quality of the underlying grids. More-over, since the training phase is performed off-line and is independent of thedifferential model the overall computational costs are kept low.Keywords:polygonal grid refinement, convolutional neural networks, virtualelement method, polygonal discontinuous Galerkin metho