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Physics-Informed Neural Network-Based Inverse Framework for Time-Fractional Differential Equations for Rheology
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
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Biology 2025, 14(7), 779; https://doi.org/10.3390/biology14070779 (registering DOI)
Submission received: 30 March 2025
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Revised: 21 June 2025
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Accepted: 26 June 2025
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Published: 27 June 2025
(This article belongs to the Special Issue Computational Modeling of Drug Delivery)
Simple Summary
Modeling complex systems with memory-dependent behavior, such as anomalous diffusion and viscoelasticity, is often hindered by limited data and the computational challenges of inverse problems. Here, we developed a Physics-Informed Neural Network (PINN) framework that incorporates time-fractional derivatives to infer key physical parameters from sparse and noisy data. Despite the recognized potential of fractional models, we found that existing PINN methods largely overlook non-integer dynamics. Using synthetic data and experimental rheology measurements, we showed that our framework could accurately recover parameters such as the generalized diffusion coefficient and fractional orders—even under 25% Gaussian noise. Notably, while traditional models require extensive parameterization, our approach predicted relaxation behavior in biological tissues with fewer assumptions and less than 10% relative error. These results suggest that while fractional PINNs offer powerful tools for learning hidden dynamics, the benefits may be constrained when noise levels are extreme or when behavioral complexity exceeds the model structure.
Abstract
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.
