This paper introduces a novel deep neural network architecture for solving the inverse scattering problem in frequency domain with wide-band data, by directly approximating the inverse map, thus avoiding the expensive optimization loop of classical methods. The architecture is motivated by the filtered back-projection formula in the full aperture regime and with homogeneous background, and it leverages the underlying equivariance of the problem and compressibility of the integral operator. This drastically reduces the number of training parameters, and therefore the computational and sample complexity of the method. In particular, we obtain an architecture whose number of parameters scale sub-linearly with respect to the dimension of the inputs, while its inference complexity scales super-linearly but with very small constants. We provide several numerical tests that show that the current approach results in better reconstruction than optimization-based techniques such as full-waveform inversion, but at a fraction of the cost while being competitive with state-of-the-art machine learning methods.

翻译：本文提出了一种新颖的深度神经网络架构，用于直接逼近逆映射，从而求解频域中利用宽频数据的逆散射问题，避免了经典方法中昂贵的优化循环。该架构的灵感来源于全孔径均匀背景下的滤波反投影公式，并利用了问题的内在等变性和积分算子的可压缩性。这极大减少了训练参数的数量，从而降低了方法的计算复杂度和样本复杂度。特别地，我们得到的架构参数数量相对于输入维度呈次线性扩展，而其推理复杂度虽呈超线性扩展，但具有极小的常数。多项数值实验表明，当前方法在重建效果上优于基于优化的技术（如全波形反演），计算成本却仅为后者的一小部分，同时与最先进的机器学习方法性能相当。