In this paper, we develop a novel data-driven approach to accelerate solving large-scale linear equation systems encountered in scientific computing and optimization. Our method utilizes self-supervised training of a graph neural network to generate an effective preconditioner tailored to the specific problem domain. By replacing conventional hand-crafted preconditioners used with the conjugate gradient method, our approach, named neural incomplete factorization (NeuralIF), significantly speeds-up convergence and computational efficiency. At the core of our method is a novel message-passing block, inspired by sparse matrix theory, that aligns with the objective to find a sparse factorization of the matrix. We evaluate our proposed method on both a synthetic and a real-world problem arising from scientific computing. Our results demonstrate that NeuralIF consistently outperforms the most common general-purpose preconditioners, including the incomplete Cholesky method, achieving competitive performance across various metrics even outside the training data distribution.

翻译：本文提出了一种新颖的数据驱动方法，用于加速科学计算与优化领域常涉及的大规模线性方程组求解。该方法利用图神经网络的自我监督训练，生成针对特定问题域的高效预处理器。通过替换传统手工设计的共轭梯度法预处理器，我们提出的神经不完全分解方法（NeuralIF）显著提升了收敛速度与计算效率。其核心是一种受稀疏矩阵理论启发的新型消息传递模块，该模块与矩阵稀疏分解目标高度契合。我们在合成问题与科学计算领域的实际问题上评估了所提方法，结果表明NeuralIF在多种指标上持续优于包括不完全Cholesky法在内的通用预处理器，即使在训练数据分布之外仍能保持竞争性表现。