MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods
This work addresses the challenge of parameter optimization in multilevel DDMs for computational science and engineering, offering a domain-specific improvement over prior methods focused on simpler or structured cases.
The paper tackles the problem of optimizing algorithmic parameters for two-level domain decomposition methods (DDMs) used in solving partial differential equations, particularly on unstructured grids, by proposing a novel multigrid graph neural network (MG-GNN) architecture with an unsupervised loss function, resulting in robust performance on problems orders of magnitude larger than training data and outperforming existing hierarchical graph networks.
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.