  Large sparse linear systems of equations are ubiquitous in science and
engineering, such as those arising from discretizations of partial differential
equations. Algebraic multigrid (AMG) methods are one of the most common methods
of solving such linear systems, with an extensive body of underlying
mathematical theory. A system of linear equations defines a graph on the set of
unknowns and each level of a multigrid solver requires the selection of an
appropriate coarse graph along with restriction and interpolation operators
that map to and from the coarse representation. The efficiency of the multigrid
solver depends critically on this selection and many selection methods have
been developed over the years. Recently, it has been demonstrated that it is
possible to directly learn the AMG interpolation and restriction operators,
given a coarse graph selection. In this paper, we consider the complementary
problem of learning to coarsen graphs for a multigrid solver, a necessary step
in developing fully learnable AMG methods. We propose a method using a
reinforcement learning (RL) agent based on graph neural networks (GNNs), which
can learn to perform graph coarsening on small planar training graphs and then
be applied to unstructured large planar graphs, assuming bounded node degree.
We demonstrate that this method can produce better coarse graphs than existing
algorithms, even as the graph size increases and other properties of the graph
are varied. We also propose an efficient inference procedure for performing
graph coarsening that results in linear time complexity in graph size.
