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

Area of Science:

• * Computational mathematics and physics.
• * Modeling of complex dynamical systems.

Background:

• * Time-fractional differential equations model memory-dependent dynamics in biotransport and viscoelasticity.
• * Solving inverse problems with these equations is challenging due to stability, uniqueness, and data limitations.
• * Existing Physics-Informed Neural Networks (PINNs) are typically limited to integer-order derivatives.

Purpose of the Study:

• * To develop a PINN framework for inverse problems with time-fractional derivatives.
• * To apply the framework to anomalous diffusion and fractional viscoelasticity.
• * To infer key physical parameters from synthetic and experimental data.

Main Methods:

• * Developed a customized PINN framework for time-fractional inverse problems.
• * Incorporated a scaled residual loss function for enhanced robustness against noise.
• * Validated the approach using synthetic datasets and experimental data from porcine tissue.

Main Results:

• * Successfully inferred generalized diffusion coefficients and fractional derivative orders in diffusion models.
• * Accurately recovered relaxation parameters in fractional Maxwell models.
• * Achieved relative errors below 10% for parameter recovery even with 25% Gaussian noise.
• * Demonstrated accurate prediction of relaxation moduli in porcine tissue experiments.

Conclusions:

• * The developed PINN framework effectively learns fractional dynamics from noisy and sparse data.
• * The approach shows significant promise for modeling complex biological and mechanical systems.
• * This work extends the applicability of PINNs to a broader class of fractional differential equations.