  Large sparse linear algebraic systems can be found in a variety of scientific
and engineering fields, and many scientists strive to solve them in an
efficient and robust manner. In this paper, we propose an interpretable neural
solver, the Fourier Neural Solver (FNS), to address them. FNS is based on deep
learning and Fast Fourier transform. Because the error between the iterative
solution and the ground truth involves a wide range of frequency modes, FNS
combines a stationary iterative method and frequency space correction to
eliminate different components of the error. Local Fourier analysis reveals
that the FNS can pick up on the error components in frequency space that are
challenging to eliminate with stationary methods. Numerical experiments on the
anisotropy diffusion equation, convection-diffusion equation, and Helmholtz
equation show that FNS is more efficient and more robust than the
state-of-the-art neural solver.
