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

Neural incomplete factorization: learning preconditioners for the conjugate gradient method