Graph Neural Multilevel Preconditioners for Iterative Solvers

2026-07-30

AI Summary

This work addresses the challenge of preconditioning indefinite or nonsymmetric sparse linear systems by proposing a novel multilevel preconditioner that integrates algebraic multigrid (AMG) hierarchies with graph neural networks (GNNs). For the first time, the method enables end-to-end learning of all-level smoothing, restriction, and interpolation operators within a unified framework, embedding AMG priors directly into the GNN architecture while remaining compatible with standard Krylov solvers. Extensive experiments on over 800 benchmark matrices demonstrate that the approach significantly accelerates convergence compared to classical AMG, ILUT, and existing GNN-based preconditioners on specific problem classes. The study also delineates the computational overhead introduced by the multilevel structure and extends the applicability of data-driven methods to non-elliptic problems.

Abstract

Solving large, sparse linear systems is a core task in scientific computing, and efficient iterative solvers rely critically on effective and robust preconditioning. While classical methods such as algebraic multigrid (AMG) are highly scalable, their robustness can degrade on indefinite or nonsymmetric systems where heuristics originally developed for elliptic PDEs are less reliable. Recently, Graph Neural Networks (GNNs) have emerged as data-driven preconditioners; yet, the practical impact of imposing an AMG-style hierarchy remains underexplored for general sparse matrices. In this work, we propose a Graph Neural Multilevel Preconditioner (GMP) that adopts an AMG hierarchy as a structural prior and learns smoothing, restriction, and interpolation operators in a unified framework. Our method targets general sparse systems and is instantiated as a drop-in preconditioner for standard Krylov solvers. On a benchmark of over 800 sparse matrices, we compare against classical AMG, single-level ILUT, and state-of-the-art GNN preconditioners, and characterize the regimes where multilevel graph neural preconditioning improves convergence or, conversely, introduces overhead relative to strong single-level baselines. These results highlight both the promise and the limitations of enforcing AMG-style multilevel structure in learned preconditioners for large-scale scientific simulations.

Problem

• preconditioning
• sparse linear systems
• graph neural networks
• algebraic multigrid
• iterative solvers

Innovation

• Graph Neural Networks
• Multilevel Preconditioning
• Algebraic Multigrid
• Iterative Solvers
• Sparse Linear Systems