Deep Learning-Enhanced Preconditioning for Efficient Conjugate Gradient Solvers in Large-Scale PDE Systems

TL;DR

This paper presents a novel GNN-based preconditioning method that significantly improves the efficiency and scalability of conjugate gradient solvers for large-scale PDE systems, validated on very large matrices.

Contribution

It introduces a GNN-integrated preconditioning approach that outperforms traditional IC, enabling scalable and efficient solutions for large PDE discretizations.

Findings

24.8% reduction in iteration counts compared to IC

Two orders of magnitude increase in training scale

Validated on matrices up to 5 million dimensions

Abstract

Preconditioning techniques are crucial for enhancing the efficiency of solving large-scale linear equation systems that arise from partial differential equation (PDE) discretization. These techniques, such as Incomplete Cholesky factorization (IC) and data-driven neural network methods, accelerate the convergence of iterative solvers like Conjugate Gradient (CG) by approximating the original matrices. This paper introduces a novel approach that integrates Graph Neural Network (GNN) with traditional IC, addressing the shortcomings of direct generation methods based on GNN and achieving significant improvements in computational efficiency and scalability. Experimental results demonstrate an average reduction in iteration counts by 24.8% compared to IC and a two-order-of-magnitude increase in training scale compared to previous methods. A three-dimensional static structural analysis utilizing finite element methods was validated on training sparse matrices of up to 5 million dimensions and inference scales of up to 10 million. Furthermore, the approach demon-strates robust generalization capabilities across scales, facilitating the effective acceleration of CG solvers for large-scale linear equations using small-scale data on modest hardware. The method's robustness and scalability make it a practical solution for computational science.