Algebraic Multigrid (AMG) is one of the most widely used iterative algorithms for solving large sparse linear equations $Ax=b$. In AMG, the coarse grid is a key component that affects the efficiency of the algorithm, the construction of which relies on the strong threshold parameter $\theta$. This parameter is generally chosen empirically, with a default value in many current AMG solvers of 0.25 for 2D problems and 0.5 for 3D problems. However, for many practical problems, the quality of the coarse grid and the efficiency of the AMG algorithm are sensitive to $\theta$; the default value is rarely optimal, and sometimes is far from it. Therefore, how to choose a better $\theta$ is an important question. In this paper, we propose a deep learning based auto-tuning method, AutoAMG($\theta$) for multiscale sparse linear equations, which are widely used in practical problems. The method uses Graph Neural Networks (GNNs) to extract matrix features, and a Multilayer Perceptron (MLP) to build the mapping between matrix features and the optimal $\theta$, which can adaptively output $\theta$ values for different matrices. Numerical experiments show that AutoAMG($\theta$) can achieve significant speedup compared to the default $\theta$ value.

翻译：代数多重网格法（AMG）是求解大规模稀疏线性方程组 $Ax=b$ 时最广泛使用的迭代算法之一。在AMG中，粗网格是影响算法效率的关键组成部分，其构建依赖于强阈值参数 $\theta$。该参数通常凭经验选择，当前许多AMG求解器中对于二维问题的默认值为0.25，三维问题为0.5。然而，对于许多实际问题，粗网格质量和AMG算法效率对 $\theta$ 高度敏感；默认值极少达到最优，有时甚至远非如此。因此，如何选择更优的 $\theta$ 是一个重要问题。本文针对实际问题中广泛使用的多尺度稀疏线性方程组，提出了一种基于深度学习的自动调优方法AutoAMG($\theta$)。该方法利用图神经网络（GNNs）提取矩阵特征，并通过多层感知机（MLP）建立矩阵特征与最优 $\theta$ 之间的映射关系，从而能够自适应地为不同矩阵输出 $\theta$ 值。数值实验表明，与默认 $\theta$ 值相比，AutoAMG($\theta$) 能够实现显著的加速效果。