  Multigrid methods are one of the most efficient techniques for solving linear
systems arising from Partial Differential Equations (PDEs) and graph Laplacians
from machine learning applications. One of the key components of multigrid is
smoothing, which aims at reducing high-frequency errors on each grid level.
However, finding optimal smoothing algorithms is problem-dependent and can
impose challenges for many problems. In this paper, we propose an efficient
adaptive framework for learning optimized smoothers from operator stencils in
the form of convolutional neural networks (CNNs). The CNNs are trained on
small-scale problems from a given type of PDEs based on a supervised loss
function derived from multigrid convergence theories, and can be applied to
large-scale problems of the same class of PDEs. Numerical results on
anisotropic rotated Laplacian problems demonstrate improved convergence rates
and solution time compared with classical hand-crafted relaxation methods.
