  In this paper, we propose Neumann Series Neural Operator (NSNO) to learn the
solution operator of Helmholtz equation from inhomogeneity coefficients and
source terms to solutions. Helmholtz equation is a crucial partial differential
equation (PDE) with applications in various scientific and engineering fields.
However, efficient solver of Helmholtz equation is still a big challenge
especially in the case of high wavenumber. Recently, deep learning has shown
great potential in solving PDEs especially in learning solution operators.
Inspired by Neumann series in Helmholtz equation, we design a novel network
architecture in which U-Net is embedded inside to capture the multiscale
feature. Extensive experiments show that the proposed NSNO significantly
outperforms the state-of-the-art FNO with at least 60\% lower relative
$L^2$-error, especially in the large wavenumber case, and has 50\% lower
computational cost and less data requirement. Moreover, NSNO can be used as the
surrogate model in inverse scattering problems. Numerical tests show that NSNO
is able to give comparable results with traditional finite difference forward
solver while the computational cost is reduced tremendously.
