Graph Neural Preconditioners for Iterative Solutions of Sparse Linear Systems

TL;DR

This paper introduces graph neural network-based preconditioners for iterative solutions of large sparse linear systems, demonstrating faster construction and execution times compared to traditional methods across diverse applications.

Contribution

It proposes a novel machine learning approach using graph neural networks as preconditioners, effective for ill-conditioned matrices and outperforming classical methods in speed.

Findings

Faster construction time than ILU and AMG preconditioners.

Reduced execution time compared to Krylov methods like GMRES.

Effective on a wide range of large, challenging matrices.

Abstract

Preconditioning is at the heart of iterative solutions of large, sparse linear systems of equations in scientific disciplines. Several algebraic approaches, which access no information beyond the matrix itself, are widely studied and used, but ill-conditioned matrices remain very challenging. We take a machine learning approach and propose using graph neural networks as a general-purpose preconditioner. They show attractive performance for many problems and can be used when the mainstream preconditioners perform poorly. Empirical evaluation on over 800 matrices suggests that the construction time of these graph neural preconditioners (GNPs) is more predictable and can be much shorter than that of other widely used ones, such as ILU and AMG, while the execution time is faster than using a Krylov method as the preconditioner, such as in inner-outer GMRES. GNPs have a strong potential for solving large-scale, challenging algebraic problems arising from not only partial differential equations, but also economics, statistics, graph, and optimization, to name a few.