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:以图神经网络为基础的多边形网格聚合方法及其在多重网格求解器中的应用
摘要:聚合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扮演着重要的角色。为了自动执行多边形网格的聚合,我们提出了使用机器学习(ML)策略,可以自然地利用网格的几何信息来保持网格质量,增强数值方法的性能,降低总体计算成本。具体来说,我们采用k-means聚类算法和图神经网络(GNN)来分割计算网格的连接图。此外,GNN具有高速的在线推理速度,并且具有同时处理网格结构和几何信息(例如元素的面积或它们的重心坐标)的优势。这些技术与METIS进行了比较,METIS是一种用于图分区的标准算法,旨在仅处理网格的图信息。我们证明了ML策略在质量指标方面的性能得到了提高,并且这种模型在应用于更复杂的几何结构(例如脑MRI扫描)时表现出了良好的泛化能力和保持网格质量的能力。在多边形不连续Galerkin(PolyDG)框架中应用于多重网格(MG)求解器时,这些策略的有效性也得到了证明。在考虑的实验中,GNN在推理速度,准确性和方法灵活性方面表现最佳。