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.

翻译：聚合技术对于降低数值模拟的计算成本至关重要，并且是多层代数求解器的基础。为了实现多面体网格的自动聚合，我们提出了一种基于几何深度学习的新型算法，该算法能够利用底层计算域的几何与物理信息来构建聚合网格，同时保证聚合网格的质量。具体而言，我们提出了一种基于图神经网络（GNN）的二分模型，用于对三维计算网格的合适连通图进行划分。该新方法具有较高的在线推理速度，能够同时处理网格的图结构、网格的几何信息（如单元体积、中心坐标）以及计算域的物理信息（如物理参数）。借助这一新方法，我们的算法能够自动完成由异质介质组成的计算域网格的聚合。我们将所提出的GNN技术与k-means算法和METIS进行了比较：后两者是标准的图划分方法，仅能处理网格的连通性信息。实验证明，我们的算法在质量指标和运行时间方面均优于现有方法。此外，我们还验证了该算法在应用于更复杂几何结构（例如从医学图像重建的三维几何）时也表现出良好的泛化能力。最后，我们在包含微观结构的问题框架中以及在人脑等复杂几何结构上，测试了该模型在异质域聚合任务中的性能。