  Algebraic Multigrid (AMG) is one of the most 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.
