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1. Identifying Arguments of Space-Time Fractional Diffusion: Data-Driven Approach

   https://doi.org/https://doi.org/10.3389/fams.2020.00014  

   Znaidi Mohamed Ridha, Gupta Gaurav (May 2020, Frontiers in applied mathematics and statistics)

   null (Ed.)

   A plethora of complex dynamical systems from disordered media to biological systems exhibit mathematical characteristics (e.g., long-range dependence, self-similar and power law magnitude increments) that are well-fitted by fractional partial differential equations (PDEs). For instance, some biological systems displaying an anomalous diffusion behavior, which is characterized by a non-linear mean-square displacement relation, can be mathematically described by fractional PDEs. In general, the PDEs represent various physical laws or rules governing complex dynamical systems. Since prior knowledge about the mathematical equations describing complex dynamical systems in biology, healthcare, disaster mitigation, transportation, or environmental sciences may not be available, we aim to provide algorithmic strategies to discover the integer or fractional PDEs and their parameters from system's evolution data. Toward deciphering non-trivial mechanisms driving a complex system, we propose a data-driven approach that estimates the parameters of a fractional PDE model. We study the space-time fractional diffusion model that describes a complex stochastic process, where the magnitude and the time increments are stable processes. Starting from limited time-series data recorded while the system is evolving, we develop a fractional-order moments-based approach to determine the parameters of a generalized fractional PDE. We formulate two optimization problems to allow us to estimate the arguments of the fractional PDE. Employing extensive simulation studies, we show that the proposed approach is effective at retrieving the relevant parameters of the space-time fractional PDE. The presented mathematical approach can be further enhanced and generalized to include additional operators that may help to identify the dominant rule governing the measurements or to determine the degree to which multiple physical laws contribute to the observed dynamics. 

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2. Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics

   https://doi.org/10.1038/s42005-024-01521-z  

   Liu, Xin-Yang; Zhu, Min; Lu, Lu; Sun, Hao; Wang, Jian-Xun (January 2024, Communications Physics)

   Abstract Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by “baking” the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers’, and Navier-Stokes equations. 

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3. Learning partial differential equations for biological transport models from noisy spatio-temporal data

   https://doi.org/10.1098/rspa.2019.0800  

   Lagergren, John H.; Nardini, John T.; Michael Lavigne, G.; Rutter, Erica M.; Flores, Kevin B. (February 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We investigate methods for learning partial differential equation (PDE) models from spatio-temporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyse the performance in using previous methods to denoise data for the task of discovering the governing system of PDEs. We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e. the advection–diffusion, classical Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) and nonlinear Fisher–KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and both local and global polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model. 

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4. Fractional Voigt-Regularization of the 3D Navier–Stokes and Euler Equations: Global Well-Posedness and Limiting Behavior

   https://doi.org/10.1007/s00021-025-00948-w  

   Brzeźniak, Zdzisław; Larios, Adam; Safarik, Isabel (August 2025, Journal of Mathematical Fluid Mechanics)

   The Voigt regularization is a technique used to model turbulent flows, offering advantages such as sharing steady states with the Navier-Stokes equations and requiring no modification of boundary conditions; however, the parabolic dissipative character of the equation is lost. In this work we propose and study a generalization of the Voigt regularization technique by introducing a fractional power $$r$$ in the Helmholtz operator, which allows for dissipation in the system, at least in the viscous case. We examine the resulting fractional Navier-Stokes-Voigt (fNSV) and fractional Euler-Voigt (fEV) and show that global well-posedness holds in the 3D periodic case for fNSV when the fractional power $$r \geq \frac{1}{2}$$ and for fEV when $$r>\frac{5}{6}$$. Moreover, we show that the solutions of these fractional Voigt-regularized systems converge to solutions of the original equations, on the corresponding time interval of existence and uniqueness of the latter, as the regularization parameter $$\alpha \to 0$$. Additionally, we prove convergence of solutions of fNSV to solutions of fEV as the viscosity $$\nu \to 0$$ as well as the convergence of solutions of fNSV to solutions of the 3D Euler equations as both $$\alpha, \nu \to 0$$. Furthermore, we derive a criterion for finite-time blow-up for each system based on this regularization. These results may be of use to researchers in both pure and applied fluid dynamics, particularly in terms of approximate models for turbulence and as tools to investigate potential blow-up of solutions. 

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5. Biologically-informed neural networks guide mechanistic modeling from sparse experimental data

   https://doi.org/10.1371/journal.pcbi.1008462  

   Lagergren, John H.; Nardini, John T.; Baker, Ruth E.; Simpson, Matthew J.; Flores, Kevin B. (December 2020, PLOS Computational Biology)

   Lavrik, Inna (Ed.)

   Biologically-informed neural networks (BINNs), an extension of physics-informed neural networks [1], are introduced and used to discover the underlying dynamics of biological systems from sparse experimental data. In the present work, BINNs are trained in a supervised learning framework to approximate in vitro cell biology assay experiments while respecting a generalized form of the governing reaction-diffusion partial differential equation (PDE). By allowing the diffusion and reaction terms to be multilayer perceptrons (MLPs), the nonlinear forms of these terms can be learned while simultaneously converging to the solution of the governing PDE. Further, the trained MLPs are used to guide the selection of biologically interpretable mechanistic forms of the PDE terms which provides new insights into the biological and physical mechanisms that govern the dynamics of the observed system. The method is evaluated on sparse real-world data from wound healing assays with varying initial cell densities [2]. 

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