Simultaneous Bayesian identification of time-dependent boundary fluxes in a time-fractional diffusion equation

In this article, we study the inverse problem of simultaneously identifying two time-dependent boundary fluxes in a one-dimensional time-fractional diffusion equation with the Caputo derivative. The fluxes appear as Neumann boundary conditions at both ends, and two temporal measurements are assumed: the spatial integral of the solution and the pointwise value at the left boundary. We prove uniqueness and conditional stability results, establishing a rigorous Hölder-type stability estimate for the boundary fluxes under suitable regularity assumptions. To address the ill-posedness, a Bayesian framework is adopted, assigning Gaussian process priors to the unknown fluxes and characterizing the posterior via Bayes’ theorem. The posterior is numerically explored using an ensemble-based sampling method, avoiding adjoint computations and showing robustness to noise. Numerical experiments confirm accurate reconstruction of both fluxes, capturing their shape and amplitude, while providing reliable uncertainty quantification (UQ).