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

NSNO: Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium