# Agglomeration of Polygonal Grids using Graph Neural Networks with applications to Multigrid solvers

arXiv: https://arxiv.org/abs/2210.17457 · HTML: https://mlsanity.com/paper/2210.17457/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 83 references that are in the map.
- First posted on arXiv 31 Oct 2022.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Graph Partitioning and Graph Neural Network based Hierarchical Graph Matching for Graph Similarity Computation](https://mlsanity.com/paper/2005.08008/) (2020)
- [Graph Partitioning and Sparse Matrix Ordering using Reinforcement Learning and Graph Neural Networks](https://mlsanity.com/paper/2104.03546/) (2021)
- [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)
- [Deep Learning and Spectral Embedding for Graph Partitioning](https://mlsanity.com/paper/2110.08614/) (2021)
- [Benchmark of Polygon Quality Metrics for Polytopal Element Methods](https://mlsanity.com/paper/1906.01627/) (2019)
- [Towards Efficient Large-Scale Graph Neural Network Computing](https://mlsanity.com/paper/1810.08403/) (2018)
- [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)

## Built on by (most cited first, 3 of 3)

- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [PolygonGNN: Representation Learning for Polygonal Geometries with Heterogeneous Visibility Graph](https://mlsanity.com/paper/2407.00742/) (2024)
- [A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs](https://mlsanity.com/paper/2304.10832/) (2023)

## Abstract (arXiv)

Agglomeration-based strategies are important both within adaptive refinement algorithms and to construct scalable multilevel algebraic solvers. In order to automatically perform agglomeration of polygonal grids, we propose the use of Machine Learning (ML) strategies, that can naturally exploit geometrical information about the mesh in order to preserve the grid quality, enhancing performance of numerical methods and reducing the overall computational cost. In particular, we employ the k-means clustering algorithm and Graph Neural Networks (GNNs) to partition the connectivity graph of a computational mesh. Moreover, GNNs have high online inference speed and the advantage to process naturally and simultaneously both the graph structure of mesh and the geometrical information, such as the areas of the elements or their barycentric coordinates. These techniques are compared with METIS, a standard algorithm for graph partitioning, which is meant to process only the graph information of the mesh. We demonstrate that performance in terms of quality metrics is enhanced for ML strategies. Such models also show a good degree of generalization when applied to more complex geometries, such as brain MRI scans, and the capability of preserving the quality of the grid. The effectiveness of these strategies is demonstrated also when applied to MultiGrid (MG) solvers in a Polygonal Discontinuous Galerkin (PolyDG) framework. In the considered experiments, GNNs show overall the best performance in terms of inference speed, accuracy and flexibility of the approach.