Physics-Informed Neural Network based inverse framework for time-fractional differential equations for rheology

TL;DR

This paper extends Physics-Informed Neural Networks to inverse problems involving time-fractional derivatives, demonstrating robustness and accuracy in modeling anomalous diffusion and viscoelasticity with noisy data.

Contribution

We develop a novel PINN framework tailored for inverse problems with fractional derivatives, incorporating a residual loss function scaled by data variability, and validate its effectiveness on complex rheological models.

Findings

Achieves less than 10% relative error in parameter estimation.

Demonstrates robustness with 25% Gaussian noise in data.

Successfully models fractional viscoelasticity and anomalous diffusion.

Abstract

Time-fractional differential equations offer a robust framework for capturing intricate phenomena characterized by memory effects, particularly in fields like biotransport and rheology. However, solving inverse problems involving fractional derivatives presents notable challenges, including issues related to stability and uniqueness. While Physics-Informed Neural Networks (PINNs) have emerged as effective tools for solving inverse problems, most existing PINN frameworks primarily focus on integer-ordered derivatives. In this study, we extend the application of PINNs to address inverse problems involving time-fractional derivatives, specifically targeting two problems: 1) anomalous diffusion and 2) fractional viscoelastic constitutive equation. Leveraging both numerically generated datasets and experimental data, we calibrate the concentration-dependent generalized diffusion coefficient and parameters for the fractional Maxwell model. We devise a tailored residual loss function that scales with the standard deviation of observed data. We rigorously test our framework's efficacy in handling anomalous diffusion. Even after introducing 25% Gaussian noise to the concentration dataset, our framework demonstrates remarkable robustness. Notably, the relative error in predicting the generalized diffusion coefficient and the order of the fractional derivative is less than 10% for all cases, underscoring the resilience and accuracy of our approach. In another test case, we predict relaxation moduli for three pig tissue samples, consistently achieving relative errors below 10%. Furthermore, our framework exhibits promise in modeling anomalous diffusion and non-linear fractional viscoelasticity.