  Large linear systems are ubiquitous in modern computational science. The main
recipe for solving them is iterative solvers with well-designed
preconditioners. Deep learning models may be used to precondition residuals
during iteration of such linear solvers as the conjugate gradient (CG) method.
Neural network models require an enormous number of parameters to approximate
well in this setup. Another approach is to take advantage of small graph neural
networks (GNNs) to construct preconditioners of the predefined sparsity
pattern. In our work, we recall well-established preconditioners from linear
algebra and use them as a starting point for training the GNN. Numerical
experiments demonstrate that our approach outperforms both classical methods
and neural network-based preconditioning. We also provide a heuristic
justification for the loss function used and validate our approach on complex
datasets.
