  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.
