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1. Applications of variable-order fractional operators: a review

   https://doi.org/10.1098/rspa.2019.0498  

   Patnaik, Sansit; Hollkamp, John P.; Semperlotti, Fabio (February 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling. 

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2. A Symmetric Neural Network to Compute Fractional Derivatives by Training with Integer Derivatives

   https://doi.org/10.1109/SII52469.2022.9708840  

   Chen, Tan; Goodwine, Bill (January 2022, 2022 IEEE/SICE International Symposium on System Integration (SII))

   Fractional calculus is an increasingly recognized important tool for modeling complicated dynamics in modern engineering systems. While, in some ways, fractional derivatives are a straight-forward generalization of integer-order derivatives that are ubiquitous in engineering modeling, in other ways the use of them requires quite a bit of mathematical expertise and familiarity with some mathematical concepts that are not in everyday use across the broad spectrum of engineering disciplines. In more colloquial terms, the learning curve is steep. While the authors recognize the need for fundamental competence in tools used in engineering, a computational tool that can provide an alternative means to compute fractional derivatives does have a useful role in engineering modeling. This paper presents the use of a symmetric neural network that is trained entirely on integer-order derivatives to provide a means to compute fractional derivatives. The training data does not contain any fractional-order derivatives at all, and is composed of only integer-order derivatives. The means by which a fractional derivative can be obtained is by requiring the neural network to be symmetric, that is, it is the composition of two identical sets of layers trained on integer-order derivatives. From that, the information contained in the nodes between the two sets of layers contains half-order derivative information 

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3. 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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4. Modeling Contacts and Hysteretic Behavior in Discrete Systems Via Variable-Order Fractional Operators

   https://doi.org/10.1115/1.4046831  

   Patnaik, Sansit; Semperlotti, Fabio (September 2020, Journal of Computational and Nonlinear Dynamics)

   Abstract The modeling of nonlinear dynamical systems subject to strong and evolving nonsmooth nonlinearities is typically approached via integer-order differential equations. In this study, we present the possible application of variable-order (VO) fractional operators to a class of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. Fractional operators are intrinsically multiscale operators that can act on both space- and time-dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or VO. In the latter case, the order can be function of either independent or state variables. We show that when using VO equations to describe the response of dynamical systems, the order can evolve as a function of the response itself; therefore, allowing a natural and seamless transition between widely dissimilar dynamics. Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. Within this context, we present a physics-driven strategy to define VO operators capable of capturing complex and evolutionary phenomena. Specific examples include hysteresis in discrete oscillators and contact problems. Despite using simplified models to illustrate the applications of VO operators, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex dynamical systems. 

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5. On the effects of memory and topology on the controllability of complex dynamical networks

   https://doi.org/10.1038/s41598-020-74269-5  

   Kyriakis, Panagiotis; Pequito, Sérgio; Bogdan, Paul (December 2020, Scientific Reports)

   null (Ed.)

   Abstract Recent advances in network science, control theory, and fractional calculus provide us with mathematical tools necessary for modeling and controlling complex dynamical networks (CDNs) that exhibit long-term memory. Selecting the minimum number of driven nodes such that the network is steered to a prescribed state is a key problem to guarantee that complex networks have a desirable behavior. Therefore, in this paper, we study the effects of long-term memory and of the topological properties on the minimum number of driven nodes and the required control energy. To this end, we introduce Gramian-based methods for optimal driven node selection for complex dynamical networks with long-term memory and by leveraging the structure of the cost function, we design a greedy algorithm to obtain near-optimal approximations in a computationally efficiently manner. We investigate how the memory and topological properties influence the control effort by considering Erdős–Rényi, Barabási–Albert and Watts–Strogatz networks whose temporal dynamics follow a fractional order state equation. We provide evidence that scale-free and small-world networks are easier to control in terms of both the number of required actuators and the average control energy. Additionally, we show how our method could be applied to control complex networks originating from the human brain and we discover that certain brain cortex regions have a stronger impact on the controllability of network than others. 

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