  Large sparse symmetric linear systems appear in several branches of science
and engineering thanks to the widespread use of the finite element method
(FEM). The fastest sparse linear solvers available implement hybrid iterative
methods. These methods are based on heuristic algorithms to permute rows and
columns or find a preconditioner matrix. In addition, they are inherently
sequential, making them unable to leverage the GPU processing power entirely.
We propose neural sparse linear solvers, a deep learning framework to learn
approximate solvers for sparse symmetric linear systems. Our method relies on
representing a sparse symmetric linear system as an undirected weighted graph.
Such graph representation is inherently permutation-equivariant and
scale-invariant, and it can become the input to a graph neural network trained
to regress the solution. We test neural sparse linear solvers on static linear
analysis problems from structural engineering. Our method is less accurate than
classic algorithms, but it is hardware-independent, fast on GPUs, and
applicable to generic sparse symmetric systems without any additional
hypothesis. Although many limitations remain, this study shows a general
approach to tackle problems involving sparse symmetric matrices using graph
neural networks.
