  Agglomeration techniques are important to reduce the computational costs of
numerical simulations and stand at the basis of multilevel algebraic solvers.
To automatically perform the agglomeration of polyhedral grids, we propose a
novel Geometrical Deep Learning-based algorithm that can exploit the
geometrical and physical information of the underlying computational domain to
construct the agglomerated grid and simultaneously guarantee the agglomerated
grid's quality. In particular, we propose a bisection model based on Graph
Neural Networks (GNNs) to partition a suitable connectivity graph of
computational three-dimensional meshes. The new approach has a high online
inference speed and can simultaneously process the graph structure of the mesh,
the geometrical information of the mesh (e.g. elements' volumes, centers'
coordinates), and the physical information of the domain (e.g. physical
parameters). Taking advantage of this new approach, our algorithm can
agglomerate meshes of a domain composed of heterogeneous media in an automatic
way. The proposed GNN techniques are compared with the k-means algorithm and
METIS: standard approaches for graph partitioning that are meant to process
only the connectivity information on the mesh. We demonstrate that the
performance of our algorithms outperforms available approaches in terms of
quality metrics and runtimes. Moreover, we demonstrate that our algorithm also
shows a good level of generalization when applied to more complex geometries,
such as three-dimensional geometries reconstructed from medical images.
Finally, the capabilities of the model in performing agglomeration of
heterogeneous domains are tested in the framework of problems containing
microstructures and on a complex geometry such as the human brain.
