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

TL;DR

The paper tackles inverse problems for time-fractional differential equations in rheology and transport, where memory effects complicate stability and uniqueness. It proposes a physics-informed neural network framework that uses automatic differentiation for spatial derivatives and finite-difference time discretization to handle Caputo-type derivatives, enabling learning of a concentration-dependent diffusion coefficient $D(c)$ and the order $\alpha$ as well as fractional Maxwell parameters. Key contributions include a physics-informed residual loss that scales with data variability, robust recovery of $D(c)$ and $\alpha$ with relative errors $<10\%$ under substantial noise, and accurate predictions of the relaxation modulus $G(t)$ for pig tissue samples. The approach offers a data-efficient path to modeling anomalous diffusion and non-linear fractional viscoelasticity with fewer parameters, and can be extended to three-dimensional problems.

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.

Figures (4)

• Figure 1: The problem set up for (a) anomalous diffusion and (b) the fractional Maxwell model, as described in section \ref{['PINNs_setup']}. Here, I denotes the identity operator, and the differential operators $\partial x$, $\partial y$ and $\partial c$ are computed using automatic differentiation.
• Figure 2: (a) The reference generalized diffusion coefficient compared to different levels of added noise. The relative errors in the (b) generalized diffusion coefficient and (c) fractional order for different levels of added noise.
• Figure 3: Plot of the experimentally obtained (a) stress and (b) strain values for a sample reported in mitra2024rheological.
• Figure 4: The predicted relaxation modulus and the experimental data corresponding to three samples of minipig skin tissue.