Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Chunlong Sun, Mengmeng Zhang, Zhidong Zhang

TL;DR

This paper tackles the inverse problem in time-domain fluorescence diffuse optical tomography (FDOT) of identifying a dynamic fluorophore source $\mu_f(x,t)$ from a single boundary measurement on $\Gamma\times(0,T)$. It proves a uniqueness result and establishes Lipschitz-type conditional stability via a weighted norm, then develops a data-driven reconstruction framework where forward and inverse problems are parameterized by deep neural networks and solved with specialized loss functions that incorporate PDE residuals and their derivatives. The authors derive generalization error estimates grounded in the stability analysis, and validate the approach with numerical experiments demonstrating robust recovery of $\mu_f$ and the associated fields $u_e$ and $u_m$ under noise. The work provides a rigorous foundation for boundary-data-driven FDOT imaging of dynamic molecular processes and offers a practical blueprint for implementing PINN-based inverse solvers with theoretical stability guarantees.

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.

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 tackles the inverse problem in time-domain fluorescence diffuse optical tomography (FDOT) of identifying a dynamic fluorophore source from a single boundary measurement on . It proves a uniqueness result and establishes Lipschitz-type conditional stability via a weighted norm, then develops a data-driven reconstruction framework where forward and inverse problems are parameterized by deep neural networks and solved with specialized loss functions that incorporate PDE residuals and their derivatives. The authors derive generalization error estimates grounded in the stability analysis, and validate the approach with numerical experiments demonstrating robust recovery of and the associated fields and under noise. The work provides a rigorous foundation for boundary-data-driven FDOT imaging of dynamic molecular processes and offers a practical blueprint for implementing PINN-based inverse solvers with theoretical stability guarantees.

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.
Paper Structure (17 sections, 15 theorems, 114 equations, 9 figures, 1 table, 2 algorithms)

This paper contains 17 sections, 15 theorems, 114 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction.
  2. Background and mathematical model.
  3. Literature and outline.
  4. Preliminaries.
  5. Eigensystem of operator $\mathcal{A}$.
  6. The auxiliary functions $\{\xi_l\}_{l=1}^\infty$ and the coefficients $\{c_{z,n}\}$.
  7. General settings of deep neural networks.
  8. The uniqueness and conditional stability.
  9. Representation of the forward operator.
  10. Conditional stability of Lipschitz type
  11. Generalization error estimates.
  12. Loss functions.
  13. The estimates of generalization error.
  14. Numerical inversions.
  15. Example 1:
  16. ...and 2 more sections

Key Result

Lemma 2.1

If $\Gamma$ is a nonempty open subset of $\partial \Omega$, then for each $n\in\mathbb N^+$, $\frac{\partial \varphi_n}{\partial{\overrightarrow{\bf n}}}$ can not vanish on $\Gamma$.

Figures (9)

  • Figure 1: The exact (upper), reconstruction results (middle) and corresponding absolute pointwise errors (bottom) for absorption coefficient $\mu_f$ at different times with noisy level $\delta=0.01$.
  • Figure 2: The reconstruction results (upper line) and corresponding absolute pointwise errors (bottom line) for absorption coefficient $\mu_f(x,y,1)$ with different noisy levels $\delta=0,\; 0.01, \;0.05,\; 0.1$.
  • Figure 3: The numerical results for $u_e$ and $u_m$ at different times $t=2/7, \;3/7, \; 5/7,\; 1$ with various noisy levels $\delta=0,\; 0.01,\; 0.1$.
  • Figure 4: The training loss (left) and the relative error for $\mu_f$ (right) after logarithmic re-scaling.
  • Figure 5: The time series relative error (test) for $\mu_f$ after logarithmic re-scaling for different noisy data.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 20 more