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

arXiv: https://arxiv.org/abs/2202.12654 · HTML: https://mlsanity.com/paper/2202.12654/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 71 references that are in the map.
- First posted on arXiv 25 Feb 2022.

## Builds on (most cited first, 4 of 4)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Refinement of polygonal grids using Convolutional Neural Networks with applications to polygonal Discontinuous Galerkin and Virtual Element methods](https://mlsanity.com/paper/2102.05738/) (2021)
- [Benchmark of Polygon Quality Metrics for Polytopal Element Methods](https://mlsanity.com/paper/1906.01627/) (2019)

## Built on by (most cited first, 2 of 2)

- [A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs](https://mlsanity.com/paper/2304.10832/) (2023)
- [Agglomeration of Polygonal Grids using Graph Neural Networks with applications to Multigrid solvers](https://mlsanity.com/paper/2210.17457/) (2022)

## Abstract (arXiv)

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.