Finding suitable preconditioners to accelerate iterative solution methods, such as the conjugate gradient method, is an active area of research. In this paper, we develop a computationally efficient data-driven approach to replace the typically hand-engineered algorithms with neural networks. Optimizing the condition number of the linear system directly is computationally infeasible. Instead, our method generates an incomplete factorization of the matrix and is, therefore, referred to as neural incomplete factorization (NeuralIF). For efficient training, we utilize a stochastic approximation of the Frobenius loss which only requires matrix-vector multiplications. At the core of our method is a novel messagepassing block, inspired by sparse matrix theory, that aligns with the objective of finding a sparse factorization of the matrix. By replacing conventional preconditioners used within the conjugate gradient method by data-driven models based on graph neural networks, we accelerate the iterative solving procedure. We evaluate our proposed method on both a synthetic and a real-world problem arising from scientific computing and show its ability to reduce the solving time while remaining computationally efficient.

翻译：寻找合适的预条件子以加速迭代求解方法（如共轭梯度法）是一个活跃的研究领域。本文提出了一种计算高效的数据驱动方法，利用神经网络替代传统手工设计的算法。直接优化线性系统的条件数在计算上不可行，因此我们的方法生成矩阵的不完全分解，并称之为神经不完全分解（NeuralIF）。为高效训练，我们采用Frobenius损失的随机近似，该近似仅需矩阵-向量乘法。方法的核心是一个受稀疏矩阵理论启发的新型消息传递模块，其目标与寻找矩阵的稀疏因子分解一致。通过用基于图神经网络的数据驱动模型替代共轭梯度法中使用的传统预条件子，我们加速了迭代求解过程。我们在合成问题与科学计算中的实际问题上评估了所提方法，结果表明该方法在保持计算效率的同时能够减少求解时间。