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

AutoAMG($\theta$): An Auto-tuned AMG Method Based on Deep Learning for Strong Threshold