Learning Preconditioner for Conjugate Gradient PDE Solvers

TL;DR

This paper introduces a machine learning-based method to generate effective preconditioners for conjugate gradient PDE solvers, improving efficiency and adaptability across different systems.

Contribution

It proposes a graph neural network approach to learn preconditioners from data, enhancing PCG solver performance for various PDEs.

Findings

Significant speed-up in solving 2D and 3D PDEs.

Preconditioners learned generalize well across different problems.

Improved convergence rates compared to traditional methods.

Abstract

Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given precision level. One challenge in PCG solvers is the selection of preconditioners, as different problem-dependent systems can benefit from different preconditioners. We present a new method to introduce \emph{inductive bias} in preconditioning conjugate gradient algorithm. Given a system matrix and a set of solution vectors arise from an underlying distribution, we train a graph neural network to obtain an approximate decomposition to the system matrix to be used as a preconditioner in the context of PCG solvers. We conduct extensive experiments to demonstrate the efficacy and generalizability of our proposed approach in solving various 2D and 3D linear second-order PDEs.