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1. End-to-End Learning Framework for Solving Non-Markovian Optimal Control

   Zhang, Xiaole; Zhang, Peiyu; Xiao, Xiongye; Li, Shixuan; Tzoumas, Vasileios; Gupta, Vijay; Bogdan, Paul (August 2025, International Conference on Machine Learning (ICML))

   Integer-order calculus fails to capture the long-range dependence (LRD) and memory effects found in many complex systems. Fractional calculus addresses these gaps through fractional-order integrals and derivatives, but fractional-order dynamical systems pose substantial challenges in system identification and optimal control tasks. In this paper, we theoretically derive the optimal control via linear quadratic regulator (LQR) for fractional-order linear time-invariant (FOLTI) systems and develop an end-to-end deep learning framework based on this theoretical foundation. Our approach establishes a rigorous mathematical model, derives analytical solutions, and incorporates deep learning to achieve data-driven optimal control of FOLTI systems. Our key contributions include: (i) proposing a novel method for system identification and optimal control strategy in FOLTI systems, (ii) developing the first end-to-end data-driven learning framework, Fractional-Order Learning for Optimal Control (FOLOC), that learns control policies from observed trajectories, and (iii) deriving theoretical bounds on the sample complexity for learning accurate control policies under fractional-order dynamics. Experimental results indicate that our method accurately approximates fractional-order system behaviors without relying on Gaussian noise assumptions, pointing to promising avenues for advanced optimal control. 

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2. 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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3. Operator-Based Uncertainty Quantification of Stochastic Fractional Partial Differential Equations

   https://doi.org/10.1115/1.4046093  

   Kharazmi, Ehsan; Zayernouri, Mohsen (December 2019, Journal of Verification, Validation and Uncertainty Quantification)

   null (Ed.)

   Abstract Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional orders as uncertain variables, we develop an operator-based uncertainty quantification framework in the context of stochastic fractional partial differential equations (SFPDEs), subject to additive random noise. We characterize different sources of uncertainty and then, propagate their associated randomness to the system response by employing a probabilistic collocation method (PCM). We develop a fast, stable, and convergent Petrov–Galerkin spectral method in the physical domain in order to formulate the forward solver in simulating each realization of random variables in the sampling procedure. 

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4. Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Beams

   https://doi.org/10.1115/1.4052286  

   Suzuki, Jorge L.; Kharazmi, Ehsan; Varghaei, Pegah; Naghibolhosseini, Maryam; Zayernouri, Mohsen (November 2021, Journal of Computational and Nonlinear Dynamics)

   Abstract Fractional models and their parameters are sensitive to intrinsic microstructural changes in anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we study the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelasticity is described by a distributed-order fractional model. We employ Hamilton's principle to obtain the equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin–Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, where the linear counterpart is numerically integrated through a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, yielding a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law decay rates, amplitude super-sensitivity at free vibration, and bifurcation in steady-state amplitude at primary resonance. 

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5. Detection of fractional quantum Hall states by entropy-sensitive measurements

   https://doi.org/10.1038/s41567-025-02813-z  

   Sultana, Nishat; Rienstra, Robert W; Watanabe, Kenji; Taniguchi, Takashi; Stroscio, Joseph A; Zhitenev, Nikolai B; Feldman, DE; Ghahari, Fereshte (March 2025, Nature Physics)

   The thermopower of a clean two-dimensional electron system is directly proportional to the entropy per charge carrier and can probe strongly interacting quantum phases such as fractional quantum Hall liquids. In particular, thermopower is a valuable parameter to probe the quasiparticle statistics that give rise to excess entropy in certain even-denominator fractional quantum Hall states. Here we demonstrate that the magneto-thermopower detection of fractional quantum Hall states is more sensitive than resistivity measurements. We do this in the context of Bernal-stacked bilayer graphene and highlight several even-denominator states at a relatively low magnetic feld. These capabilities of thermopower measurements support the interest in fractional quantum Hall states for fnding quasiparticles with non-Abelian statistics and elevate bilayer graphene as a promising platform for achieving this. 

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