In this paper, we consider a deep learning approach to the limited aperture inverse obstacle scattering problem. It is well known that traditional deep learning relies solely on data, which may limit its performance for the inverse problem when only indirect observation data and a physical model are available. A fundamental question arises in light of these limitations: is it possible to enable deep learning to work on inverse problems without labeled data and to be aware of what it is learning? This work proposes a deep decomposition method (DDM) for such purposes, which does not require ground truth labels. It accomplishes this by providing physical operators associated with the scattering model to the neural network architecture. Additionally, a deep learning based data completion scheme is implemented in DDM to prevent distorting the solution of the inverse problem for limited aperture data. Furthermore, apart from addressing the ill-posedness imposed by the inverse problem itself, DDM is the first physics-aware machine learning technique that can have interpretability property for the obstacle detection. The convergence result of DDM is theoretically investigated. We also prove that adding small noise to the input limited aperture data can introduce additional regularization terms and effectively improve the smoothness of the learned inverse operator. Numerical experiments are presented to demonstrate the validity of the proposed DDM even when the incident and observation apertures are extremely limited.

翻译：本文研究一种针对有限孔径逆障碍散射问题的深度学习方法。众所周知，传统深度学习完全依赖数据，当仅能获得间接观测数据和物理模型时，其在逆问题中的性能可能受限。基于这些局限性，一个根本性问题随之产生：能否使深度学习在没有标注数据的情况下处理逆问题，并使其理解自身所学内容？为此，本研究提出一种无需真实标签的深度分解方法（DDM）。该方法通过向神经网络架构提供与散射模型相关的物理算子来实现这一目标。此外，DDM中实施了基于深度学习的数据补全方案，以防止有限孔径数据对逆问题解的扭曲。值得注意的是，除了处理逆问题本身带来的不适定性之外，DDM是首个具备障碍物检测可解释性特性的物理感知机器学习技术。本文从理论上研究了DDM的收敛性结果，并证明在输入有限孔径数据中添加微小噪声能够引入额外的正则化项，有效提升所学逆算子的平滑性。数值实验表明，即使在入射与观测孔径极其有限的情况下，所提出的DDM方法仍具有有效性。