  This paper studies deep neural networks for solving extremely large linear
systems arising from highdimensional problems. Because of the curse of
dimensionality, it is expensive to store both the solution and right-hand side
vector in such extremely large linear systems. Our idea is to employ a neural
network to characterize the solution with much fewer parameters than the size
of the solution under a matrix-free setting. We present an error analysis of
the proposed method, indicating that the solution error is bounded by the
condition number of the matrix and the neural network approximation error.
Several numerical examples from partial differential equations, queueing
problems, and probabilistic Boolean networks are presented to demonstrate that
the solutions of linear systems can be learned quite accurately.
