MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain
  Decomposition Methods
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.
Ali Taghibakhshi, Nicolas Nytko, Tareq Uz Zaman, Scott MacLachlan, Luke Olson, Matthew West
# MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods

###### Abstract

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.

Machine Learning, ICML

## 1 Introduction

Differential equations are at the core of many important scientific and engineering problems (Gholizadeh et al., 2021; Han and Lin, 2011), and often, there is no analytical solution available; hence, researchers utilize numerical solvers (Vos et al., 2002; Gholizadeh et al., 2023). Among numerical methods for solving the systems of equations obtained from discretization of partial differential equations (PDEs), domain decomposition methods (DDMs) are a popular approach (Toselli and Widlund, 2005; Quarteroni and Valli, 1999; Dolean et al., 2015). They have been extensively studied and applied to elliptic boundary value problems, but are also considered for time-dependent problems. Schwarz methods are among the simplest and most popular types of DDM, and map well to MPI-style parallelism, with both one-level and multilevel variants. One-level methods decompose the global problem into multiple subproblems (subdomains), which are obtained either by discretizing the same PDE over a physical subdomain or by projection onto a discrete basis, using subproblem solutions to form a preconditioner for the global problem. Classical Schwarz methods generally consider Dirichlet or Neumann boundary conditions between the subdomains, while Optimized Schwarz methods (OSM) (Gander et al., 2000) consider a combination of Dirichlet and Neumann boundary conditions, known as Robin-type boundary conditions, to improve the convergence of the method. Restricted additive Schwarz (RAS) methods (Cai and Sarkis, 1999) are a common form of Schwarz methods, and optimized versions of one-level RAS has been theoretically studied by St-Cyr et al. (2007). Two-level methods extend one-level approaches by adding a (global) coarse-grid correction step to the preconditioner, generally improving performance but at an added cost.

In recent years, there has been a growing focus on using machine learning (ML) methods to learn optimized parameters for iterative PDE solvers, including DDM and algebraic multigrid (AMG). In Greenfeld et al. (2019) convolutional neural networks (CNNs) are used to learn the interpolation operator in AMG on structured problems, and in a following study (Luz et al., 2020), graph neural networks (GNNs) are used to extend the results to arbitrary unstructured grids. In a different fashion, reinforcement learning methods along with GNNs are used to learn coarse-gird selection in reduction-based AMG in Taghibakhshi et al. (2021). As mentioned in Heinlein et al. (2021), when combining ML methods with DDM, approaches can be categorized into two main families, namely using ML within a classical DDM framework to obtain improved convergence and using deep neural networks as the main solver or discretization module for DDMs. In a recent study (Taghibakhshi et al., 2022), GNNs are used to learn interface conditions