  Efficient numerical solvers for sparse linear systems are crucial in science
and engineering. One of the fastest methods for solving large-scale sparse
linear systems is algebraic multigrid (AMG). The main challenge in the
construction of AMG algorithms is the selection of the prolongation operator --
a problem-dependent sparse matrix which governs the multiscale hierarchy of the
solver and is critical to its efficiency. Over many years, numerous methods
have been developed for this task, and yet there is no known single right
answer except in very special cases. Here we propose a framework for learning
AMG prolongation operators for linear systems with sparse symmetric positive
(semi-) definite matrices. We train a single graph neural network to learn a
mapping from an entire class of such matrices to prolongation operators, using
an efficient unsupervised loss function. Experiments on a broad class of
problems demonstrate improved convergence rates compared to classical AMG,
demonstrating the potential utility of neural networks for developing sparse
system solvers.
