  Constructing fast numerical solvers for partial differential equations (PDEs)
is crucial for many scientific disciplines. A leading technique for solving
large-scale PDEs is using multigrid methods. At the core of a multigrid solver
is the prolongation matrix, which relates between different scales of the
problem. This matrix is strongly problem-dependent, and its optimal
construction is critical to the efficiency of the solver. In practice, however,
devising multigrid algorithms for new problems often poses formidable
challenges. In this paper we propose a framework for learning multigrid
solvers. Our method learns a (single) mapping from a family of parameterized
PDEs to prolongation operators. We train a neural network once for the entire
class of PDEs, using an efficient and unsupervised loss function. Experiments
on a broad class of 2D diffusion problems demonstrate improved convergence
rates compared to the widely used Black-Box multigrid scheme, suggesting that
our method successfully learned rules for constructing prolongation matrices.
