Polytopal mesh agglomeration via geometrical deep learning for three-dimensional heterogeneous domains

Agglomeration techniques can be successfully employed to reduce the computational costs of numerical simulations and stand at the basis of multilevel algebraic solvers. To automatically perform mesh agglomeration, 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. It 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, automatically respecting the underlying heterogeneities. The proposed GNN approach is compared with the k-means algorithm and METIS, which are widely employed approaches for graph partitioning and are meant to process only the connectivity information on the mesh. We demonstrate that the performance of our algorithms outperforms the k-means and METIS algorithms in terms of quality metrics and runtimes. Moreover, we demonstrate that our algorithm also shows a good level of generalization when applied to complex geometries, such as three-dimensional geometries reconstructed from medical images. Finally, the model's capability to perform agglomeration in heterogeneous domains is evaluated when integrated into a polytopal discontinuous Galerkin finite element solver.