用图神经网络学着直接生成稀疏预处理矩阵，加快线性方程求解。
Learning incomplete factorization preconditioners for GMRES
- 用图神经网络替代手工设计的不完全分解算法
- 在合成数据上减少GMRES迭代次数并改善谱性质
- 输出自动稀疏且可逆，适合大规模稀疏矩阵求解
稀疏矩阵的不完全LU分解广泛用于克里洛夫子空间方法中作为预处理器以加速线性系统求解。然而，预处理器的计算过程耗时且对超参数敏感。为此，我们用图神经网络替代手工设计的算法，直接学习矩阵分解。为使神经网络输出可用作预处理器，提出一种保证预测分解可逆的输出激活函数。此外，图神经网络结构确保输出本身稀疏，具有计算优势。我们从理论上分析并实验评估了不同损失函数训练学习型预处理器的效果，在合成数据上验证其能有效减少GMRES迭代次数并改善谱性质。代码已公开于https://github.com/paulhausner/neural-incomplete-factorization。
原文摘要 · Abstract (English)
Incomplete LU factorizations of sparse matrices are widely used as preconditioners in Krylov subspace methods to speed up solving linear systems. Unfortunately, computing the preconditioner itself can be time-consuming and sensitive to hyper-parameters. Instead, we replace the hand-engineered algorithm with a graph neural network that is trained to approximate the matrix factorization directly. To apply the output of the neural network as a preconditioner, we propose an output activation function that guarantees that the predicted factorization is invertible. Further, applying a graph neural network architecture allows us to ensure that the output itself is sparse which is desirable from a computational standpoint. We theoretically analyze and empirically evaluate different loss functions to train the learned preconditioners and show their effectiveness in decreasing the number of GMRES iterations and improving the spectral properties on synthetic data. The code is available at https://github.com/paulhausner/neural-incomplete-factorization.
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