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

We study the inverse identification of a time-dependent reaction coefficient in a space-time fractional diffusion equation from integral-in-space measurements. We establish well-posedness of the direct problem, identifiability of the coefficient under a mild non-degeneracy condition, and a local conditional stability estimate. Two reconstruction methods, a conjugate gradient method and a physics-informed neural network, are developed and compared. Numerical experiments with smooth and nonsmooth coefficients under two noise levels confirm stable recovery and highlight the complementary strengths of both approaches.