  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.
