Machine Learning-Enhanced Refinement and Agglomeration Strategies for Polygonal and Polyhedral Methods

In this talk we discuss how to enhance the accuracy and performance of Polyhedral Finite Element Methods based on designing suitable Machine Learning-aided numerical algorithms to handle the process of grid refinement and agglomeration. More specifically, we propose new strategies to handle polytopal grid refinement, to be employed within an adaptive framework. Specifically, Convolutional Neural Networks are employed to classify the “shape” of an element so as to apply “ad-hoc” refinement criteria or to enhance existing refinement strategies at a low online computational cost. We test the proposed algorithms considering two families of finite element methods that support arbitrarily shaped polytopal elements, namely the Virtual Element method and the Polytopal Discontinuous Galerkin method. In the second part of the talk ML-aided grid agglomeration techniques are presented. Mesh agglomeration strategies are important both within adaptive refinement algorithms and to construct multilevel algebraic solvers. We propose to use Graph Neural Networks (GNNs) to automatically perform grid agglomeration. GNNs have the advantage to process naturally and simultaneously both the graph structure of mesh and the geometrical information. We assess the performance of the proposed agglomeration algorithm and demonstrate its effectiveness when employed within multigrid solvers in a Polytopal Discontinuous Galerkin framework.