This work considers the inverse dynamic source problem arising from the time-domain fluorescence diffuse optical tomography (FDOT). We recover the dynamic distributions of fluorophores in biological tissue by the one single boundary measurement in finite time domain. We build the uniqueness theorem of this inverse problem. After that, we introduce a weighted norm and establish the conditional stability of Lipschitz type for the inverse problem by this weighted norm. The numerical inversions are considered under the framework of the deep neural networks (DNNs). We establish the generalization error estimates rigorously derived from Lipschitz conditional stability of inverse problem. Finally, we propose the reconstruction algorithms and give several numerical examples illustrating the performance of the proposed inversion schemes.

翻译：本文研究时域荧光扩散光学断层成像（FDOT）中的逆动态源问题。我们通过有限时间域内的单个边界测量恢复生物组织中荧光团的动态分布。首先建立了该逆问题的唯一性定理。随后引入加权范数，并基于该加权范数建立了逆问题的Lipschitz型条件稳定性。数值反演在深度神经网络（DNNs）框架下进行，并严格推导了基于逆问题Lipschitz条件稳定性的泛化误差估计。最后，提出了重建算法，并通过数值算例展示了所提反演方案的有效性。