Recently, deep neural networks (DNNs) have become powerful tools for solving inverse scattering problems. However, the approximation and generalization rates of DNNs for solving these problems remain largely under-explored. In this work, we introduce two types of combined DNNs (uncompressed and compressed) to reconstruct two coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An analysis of the approximation and generalization capabilities of the proposed neural networks for simulating the regularized pseudo-inverses of the linearized forward operators in direct scattering problems is provided. The results show that, with sufficient training data and parameters, the proposed neural networks can effectively approximate the inverse process with desirable generalization. Preliminary numerical results show the feasibility of the proposed neural networks for recovering two types of isotropic inhomogeneous media. Furthermore, the trained neural network is capable of reconstructing the isotropic representation of certain types of anisotropic media.

翻译：近年来，深度神经网络已成为求解逆散射问题的有力工具。然而，深度神经网络在求解此类问题时的逼近能力与泛化性能仍鲜有研究。本文针对从两个不同频率的散射数据中重构亥姆霍兹方程中两个系数的逆散射问题，提出了两种组合式深度神经网络结构（未压缩型与压缩型）。我们分析了所提神经网络在模拟正散射问题中线性化前向算子正则化伪逆时的逼近与泛化能力。结果表明，在充足的训练数据与网络参数条件下，所提神经网络能够有效逼近逆过程并具备良好的泛化性能。初步数值实验验证了所提神经网络在重构两类各向同性非均匀介质中的可行性。此外，训练后的神经网络还能够重构特定类型各向异性介质的各向同性等效表征。