Conditional well-posedness and data-driven method for identifying the dynamic source in a coupled diffusion system from one single boundary measurement

TL;DR

This paper addresses the inverse problem of recovering dynamic fluorophore distributions in tissue from a single boundary measurement, establishing theoretical uniqueness and stability, and proposing DNN-based reconstruction algorithms with numerical validation.

Contribution

It introduces a new theoretical framework for the inverse problem in FDOT, including a uniqueness theorem and conditional stability, and develops data-driven neural network methods for practical reconstruction.

Findings

Proved uniqueness of the inverse problem.

Established Lipschitz-type conditional stability.

Demonstrated effective numerical reconstructions.

Abstract

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.