Physics-Informed Neural Network-Based Inverse Framework for Time-Fractional Differential Equations for Rheology.

Inverse problems involving time-fractional differential equations have become increasingly important for modeling systems with memory-dependent dynamics, particularly in biotransport and viscoelastic materials. Despite their potential, these problems remain challenging due to issues of stability, non-uniqueness, and limited data availability. Recent advancements in Physics-Informed Neural Networks (PINNs) offer a data-efficient framework for solving such inverse problems, yet most implementations are restricted to integer-order derivatives. In this work, we develop a PINN-based framework tailored for inverse problems involving time-fractional derivatives. We consider two representative applications: anomalous diffusion and fractional viscoelasticity. Using both synthetic and experimental datasets, we infer key physical parameters including the generalized diffusion coefficient and the fractional derivative order in the diffusion model and the relaxation parameters in a fractional Maxwell model. Our approach incorporates a customized residual loss function scaled by the standard deviation of observed data to enhance robustness. Even under 25% Gaussian noise, our method recovers model parameters with relative errors below 10%. Additionally, the framework accurately predicts relaxation moduli in porcine tissue experiments, achieving similar error margins. These results demonstrate the framework's effectiveness in learning fractional dynamics from noisy and sparse data, paving the way for broader applications in complex biological and mechanical systems.