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（PDEs）离散化系统的流行求解器，包括单层和多层变体。这些求解器依赖于多个算法和数学参数，规定了重叠区域、子域边界条件以及DDM的其他特性。尽管已有研究对优化这些参数进行了探索，但主要集中在单层设置或特定情况（如具有规则子域构造的结构化网格离散化）上。本文提出多重网格图神经网络（MG-GNN），一种用于优化双层DDM参数的新型GNN架构。我们采用一种新的无监督损失函数训练MG-GNN，使其能够在小型问题上进行有效训练，并在比训练集大数个数量级的非结构化网格上展现出鲁棒性能。研究证明，MG-GNN在此优化任务中优于流行的分层图网络架构，且所提出的损失函数对实现这一性能提升至关重要。