Study Compares Methods for Identifying Coefficients in Fractional Diffusion Equations

In research published in the journal Boundary Value Problems, author Eman Alruwaili reported on the inverse identification of a time-dependent reaction coefficient within a space-time fractional diffusion equation. The research investigated the recovery of the time-dependent parameter using integral-in-space measurements. In addressing the mathematical formulation of the system, the study first analyzed the direct problem associated with the fractional equation, establishing its formal well-posedness as the foundational framework for the inverse investigation.

The mathematical analysis focused on the conditions required to determine the unknown coefficient uniquely from space-integral data. Alruwaili established the identifiability of the time-dependent reaction coefficient under a mild non-degeneracy condition. Furthermore, the work established a local conditional stability estimate for the inverse problem. This theoretical stability result provides mathematical bounds governing the inverse identification of the reaction coefficient when reconstructing it from integral measurements under the specified non-degeneracy requirement.

To address the practical recovery of the time-dependent reaction coefficient, the author developed and compared two distinct numerical reconstruction methods. One reconstruction strategy was formulated using a conjugate gradient method, while the second strategy was developed using a physics-informed neural network framework, designated as a gPINN approach. The study designed both the conjugate gradient algorithm and the neural network approach to recover the unknown reaction coefficient from the integral-in-space measurement data.

The comparative performance of the conjugate gradient method and the physics-informed neural network was evaluated through numerical experiments. These computational tests examined the reconstruction of both smooth coefficients and nonsmooth coefficients to evaluate the behavior of each approach across different coefficient profiles. To assess recovery performance under imperfect data conditions, the author conducted the numerical experiments under two distinct noise levels.

The numerical experiments confirmed the stable recovery of the time-dependent reaction coefficient across the tested scenarios. Both the conjugate gradient method and the physics-informed neural network demonstrated stability when reconstructing smooth and nonsmooth coefficients subject to the two noise levels. In evaluating the comparative outcomes, the study highlighted the complementary strengths of the conjugate gradient approach and the gPINN approach for the inverse identification problem.

Subject of Research: Inverse identification of parameters in fractional diffusion equations

Article Title: Inverse identification of a time-dependent coefficient in a fractional diffusion equation from integral measurements: a comparative study of conjugate gradient and gPINN approaches

Article References: Alruwaili, E. (2026). Inverse identification of a time-dependent coefficient in a fractional diffusion equation from integral measurements: a comparative study of conjugate gradient and gPINN approaches. Boundary Value Problems. https://doi.org/10.1186/s13661-026-02383-w

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DOI: 10.1186/s13661-026-02383-w

Keywords: fractional diffusion equation, inverse problem, parameter identification, conjugate gradient method, physics-informed neural networks, gPINN, integral measurements, applied mathematics, Inverse, identification, time-dependent, coefficient