This paper is concerned with the inverse problem of reconstructing an inhomogeneous medium from the acoustic far-field data at a fixed frequency in two dimensions. This inverse problem is severely ill-posed (and also strongly nonlinear), and certain regularization strategy is thus needed. However, it is difficult to select an appropriate regularization strategy which should enforce some a priori information of the unknown scatterer. To address this issue, we plan to use a deep learning approach to learn some a priori information of the unknown scatterer from certain ground truth data, which is then combined with a traditional iteration method to solve the inverse problem. Specifically, we propose a deep learning-based iterative reconstruction algorithm for the inverse problem, based on a repeated application of a deep neural network and the iteratively regularized Gauss-Newton method (IRGNM). Our deep neural network (called the learned projector in this paper) mainly focuses on learning the a priori information of the shape of the unknown contrast with a normalization technique in the training process and is trained to act like a projector which is helpful for projecting the solution into some feasible region. Extensive numerical experiments show that our reconstruction algorithm provides good reconstruction results even for the high contrast case and has a satisfactory generalization ability.

翻译：本文研究利用固定频率声波远场数据在二维空间中重建非均匀介质的反问题。该反问题高度病态（且强非线性），因此需要适当的正则化策略。然而，由于难以选择既能强制施加未知散射体先验信息又合适的正则化策略，我们计划采用深度学习方法从特定真实数据中学习未知散射体的先验信息，并将其与传统迭代方法结合求解该反问题。具体而言，我们提出基于深度学习的迭代重建算法，该算法通过深度神经网络与迭代正则化高斯-牛顿法（IRGNM）的重复应用实现。本文提出的深度神经网络（称为学习投影算子）在训练过程中利用归一化技术重点学习未知对比度形状的先验信息，其被训练为类似投影算子的形式，有助于将解投影至可行域。大量数值实验表明，即使在高对比度情况下，该重建算法仍能获得良好重建结果，并具有令人满意的泛化能力。