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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
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Newtonian fluids exhibit a constant viscosity, meaning their shear stress and shear strain rate are directly proportional. This property ensures a predictable and stable response to applied forces, maintaining a linear relationship between force and flow. Examples include water, air, and light oils, consistently demonstrating this proportional behavior regardless of external conditions.
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James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and...
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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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Updated: Sep 13, 2025

Combining Microfluidics and Microrheology to Determine Rheological Properties of Soft Matter during Repeated Phase Transitions
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Physics-Informed Neural Network-Based Inverse Framework for Time-Fractional Differential Equations for Rheology.

Sukirt Thakur1, Harsa Mitra1, Arezoo M Ardekani1

  • 1School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA.

Biology
|July 29, 2025
PubMed
Summary

Physics-Informed Neural Networks (PINNs) now solve inverse problems with time-fractional derivatives. This data-efficient method accurately models complex systems like anomalous diffusion and viscoelasticity, even with noisy data.

Keywords:
anomalous diffusionfractional modelingphysics-informed machine learningrheology

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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.