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.

翻译：本文提出诺伊曼级数神经算子（NSNO），用于学习从非均匀性系数和源项到亥姆霍兹方程解的算子。亥姆霍兹方程是一类重要的偏微分方程，在科学与工程领域具有广泛应用。然而，高效求解亥姆霍兹方程仍是一大挑战，尤其是在高波数情形下。近年来，深度学习在求解偏微分方程，特别是学习解算子方面展现出巨大潜力。受亥姆霍兹方程中诺伊曼级数的启发，我们设计了一种新颖的网络架构，其中嵌入U-Net以捕获多尺度特征。大量实验表明，所提出的NSNO显著优于当前最先进的FNO，在相对$L^2$误差上至少降低60%，尤其在高波数情形下表现突出，同时计算成本降低50%且数据需求更少。此外，NSNO可作为逆散射问题中的替代模型。数值测试显示，NSNO能够给出与传统有限差分正演求解器相媲美的结果，而计算成本大幅降低。