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.

翻译：基于聚集的策略在自适应细化算法和构建可扩展的多层代数求解器中均具有重要意义。为了自动实现多边形网格的聚集，我们提出采用机器学习策略，该策略能够自然地利用网格的几何信息来保持网格质量，从而提升数值方法性能并降低整体计算成本。具体而言，我们采用k均值聚类算法和图神经网络对计算网格的连通图进行划分。此外，图神经网络具有高在线推理速度的优势，并且能够同时自然地处理网格的图结构及其几何信息（如单元面积或重心坐标）。这些技术将与标准图划分算法METIS进行比较——该算法仅处理网格的图信息。我们证明，在质量指标方面，机器学习策略可提升性能。当应用于更复杂几何体（如脑部MRI扫描）时，这些模型还展现出良好的泛化能力，并能保持网格质量。在应用于多边形不连续伽辽金框架中的多重网格求解器时，这些策略的有效性也得到了验证。在实验中，图神经网络在推理速度、精度和灵活性方面整体表现最佳。