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

arXiv: https://arxiv.org/abs/2102.05738 · HTML: https://mlsanity.com/paper/2102.05738/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 54 references that are in the map.
- First posted on arXiv 10 Feb 2021.

## Builds on (most cited first, 3 of 3)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Benchmark of Polygon Quality Metrics for Polytopal Element Methods](https://mlsanity.com/paper/1906.01627/) (2019)

## Built on by (most cited first, 5 of 5)

- [Learning robust marking policies for adaptive mesh refinement](https://mlsanity.com/paper/2207.06339/) (2022)
- [Quasi-optimal hp-finite element refinements towards singularities via deep neural network prediction](https://mlsanity.com/paper/2209.05844/) (2022)
- [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)
- [Machine Learning based refinement strategies for polyhedral grids with applications to Virtual Element and polyhedral Discontinuous Galerkin methods](https://mlsanity.com/paper/2202.12654/) (2022)

## Abstract (arXiv)

We propose new strategies to handle polygonal grids refinement based on Convolutional Neural Networks (CNNs). We show that CNNs can be successfully employed to identify correctly the "shape" of a polygonal element so as to design suitable refinement criteria to be possibly employed within adaptive refinement strategies. We propose two refinement strategies that exploit the use of CNNs to classify elements' shape, at a low computational cost. We test the proposed idea considering two families of finite element methods that support arbitrarily shaped polygonal elements, namely Polygonal Discontinuous Galerkin (PolyDG) methods and Virtual Element Methods (VEMs). We demonstrate that the proposed algorithms can greatly improve the performance of the discretization schemes both in terms of accuracy and quality of the underlying grids. Moreover, since the training phase is performed off-line and is independent of the differential model the overall computational costs are kept low.