Domain decomposition methods (DDMs) are popular solvers for discretized systems of partial differential equations (PDEs), with one-level and multilevel variants. These solvers rely on several algorithmic and mathematical parameters, prescribing overlap, subdomain boundary conditions, and other properties of the DDM. While some work has been done on optimizing these parameters, it has mostly focused on the one-level setting or special cases such as structured-grid discretizations with regular subdomain construction. In this paper, we propose multigrid graph neural networks (MG-GNN), a novel GNN architecture for learning optimized parameters in two-level DDMs\@. We train MG-GNN using a new unsupervised loss function, enabling effective training on small problems that yields robust performance on unstructured grids that are orders of magnitude larger than those in the training set. We show that MG-GNN outperforms popular hierarchical graph network architectures for this optimization and that our proposed loss function is critical to achieving this improved performance.

翻译：区域分解方法（DDMs）是求解偏微分方程离散系统的常用求解器，包含单层与多层变体。该类求解器依赖多个算法与数学参数，用于指定重叠区域、子域边界条件及DDM的其他性质。尽管已有研究致力于优化这些参数，但主要集中于单层设定或规则子域构造的结构化网格离散化等特殊情况。本文提出多重网格图神经网络（MG-GNN）——一种用于优化两层DDM参数的新型图神经网络架构。我们采用新型无监督损失函数训练MG-GNN，使其能够在小规模问题上高效训练，并对规模远超训练集的非结构化网格展现出稳健性能。实验表明，在针对该优化任务时，MG-GNN优于主流层次化图网络架构，而本文提出的损失函数对实现这种性能提升至关重要。