Machine Learning based refinement strategies for polyhedral grids with applications to Virtual Element and polyhedral Discontinuous Galerkin methods

TL;DR

This paper introduces machine learning-based strategies for polyhedral grid refinement, utilizing clustering and neural networks to improve mesh quality and computational efficiency in finite element methods.

Contribution

It presents novel ML-driven refinement strategies, including k-means clustering and CNN classification, tailored for polyhedral mesh adaptation in VEM and PolyDG methods.

Findings

Strategies preserve grid structure and quality

Reduce computational cost and mesh complexity

Effective for arbitrary polyhedral elements

Abstract

We propose two new strategies based on Machine Learning techniques to handle polyhedral grid refinement, to be possibly employed within an adaptive framework. The first one employs the k-means clustering algorithm to partition the points of the polyhedron to be refined. This strategy is a variation of the well known Centroidal Voronoi Tessellation. The second one employs Convolutional Neural Networks to classify the "shape" of an element so that "ad-hoc" refinement criteria can be defined. This strategy can be used to enhance existing refinement strategies, including the k-means strategy, at a low online computational cost. We test the proposed algorithms considering two families of finite element methods that support arbitrarily shaped polyhedral elements, namely the Virtual Element Method (VEM) and the Polygonal Discontinuous Galerkin (PolyDG) method. We demonstrate that these strategies do preserve the structure and the quality of the underlaying grids, reducing the overall computational cost and mesh complexity.