Search Results (154)

Computational modeling and machine learning have impacted several different areas of science and accelerated advancements in multiple venues. Yet traditional machine learning models have many well-known drawbacks: besides demanding a significant amount of data, they may fail to generalize beyond training data, are often treated as “black boxes”, and may predict physically inconsistent results. In response to these limitations, Physics-Informed Machine Learning (PIML) has emerged as a new area that integrates domain knowledge, such as energy or mass conservation, directly into data-driven algorithms. This review paper examines the foundations and main strategies of PIML, organizing the approaches into three categories: automated discovery and system identification, continuous-time modeling, and operator learning. In addition, Physics-Informed Neural Networks are analyzed in a dedicated section that covers architecture fundamentals, forward and inverse problem formulations, loss function design and implementation challenges. The paper also presents a survey of interdisciplinary applications of PIML in materials science, biomedical engineering, and fractional calculus. In this context, the review also analyzes open challenges and outlines future directions in the field. Full article

Dynamic knowledge representation in curved manifolds conventionally relies on integer-order Markovian sequence encoders, intrinsically yielding exponential memory decay. This paradigm fails to model the anomalous diffusion and heavy-tailed historical dependencies inherent in complex evolutionary networks and dense physical environments. This manuscript proposes the Fractional–Temporal Lorentz Graph Convolutional Network (FTL-GCN), formalizing temporal evolution as a continuous fractional geometric flow explicitly defined on the tangent bundle of the Lorentz manifold. Analytical derivations demonstrate that the discrete Grünwald–Letnikov memory kernel establishes a non-exponential, power-law lower bound for historical state retention, preventing topological manifold collapse over extended temporal horizons. Empirical evaluations demonstrate that FTL-GCN achieves competitive forecasting accuracy against the latest 2025–2026 state-of-the-art discrete models within specific temporal windows, while uniquely mitigating predictive degradation by up to 52% in long-horizon dependency stress tests and maintaining sub-millisecond latency for physical control. The architecture is subsequently deployed within an in silico biophysical simulation for autonomous micro–nano robotic navigation in the Tumor Microenvironment (TME). By establishing a physical-mathematical structural analogy—mapping the empirical fractional viscoelasticity of the extracellular matrix to the cognitive network’s fractional derivative order—FTL-GCN sustains continuous-space navigation policies in dense anomalous environments where standard integer-order models experience mechanical slip. Full article

This research paper demonstrates how to manage Caputo fractional neutral integro-differential equations which include both integral and nonlinear elements through a unified framework that models dynamic systems with memory-based dynamics. The research establishes sufficient conditions for controllability through fixed point theory in a Banach space framework which requires particular assumptions while the study focuses on the ${\mathfrak{K}}_1<1$ condition which leads to the existence of a controllable solution. The proposed criteria are demonstrated through a numerical example which tests the theoretical results. The real-world case study uses artificial neural network (ANN) technology to predict Litecoin prices through the application of the fractional controllability model which analyzes historical financial data. The hybrid framework enables precise forecasting of nonlinear time series because it combines fractional calculus mathematical principles with ANN learning abilities. The proposed method demonstrates its predictive efficiency. The method shows robust performance through experimental results using cross-validation and performance metrics. The proposed model demonstrates competitive performance while providing additional advantages such as incorporation of memory effects and theoretical controllability. The research establishes a novel connection between fractional dynamical systems and machine learning which serves as an essential tool for studying complicated systems in theoretical research and practical applications. Full article

Polarimetric descattering imaging has attracted growing interest due to its fundamental physical significance and potential applications. While deep learning has accelerated its development through powerful feature extraction and inference capabilities, existing methods still face limitations in practical scenarios, particularly under dynamic non-uniform scattering conditions such as cement dust environments. To address this, we propose a deep neural network based on the Mueller matrix model that effectively integrates polarization evolution information with deep learning. Specifically, local concentrations of the scattering medium in non-uniform cement dust are characterized by the evolution of the degree of linear polarization (DoLP), which is converted into pixel-wise weight biases to generate customized Mueller matrices adaptable to varying concentrations. The network predicts a pixel-wise dust concentration map and applies the corresponding concentration-specific Mueller matrix to each pixel for polarization-aware dehazing, ensuring physical consistency with Mueller matrix calculus throughout inference. This framework is further enhanced by a physics-constrained optimization loss and multi-scale feature fusion. Experimental results demonstrate the method’s effectiveness and superiority in diverse dynamic non-uniform cement dust environments. Full article

This article focuses on the fixed-time synchronization problem for fractional-order fuzzy cellular neural networks (FOFCNNs) with information interactions and time-varying delays. To capture the complex dynamics of practical networks, nonlinear activation functions along with fuzzy AND and OR operators are incorporated into the master–slave systems. To achieve fixed-time synchronization despite these complexities, a novel adaptive multi-module controller is proposed. This controller integrates three functionally distinct components to accelerate the convergence rate, eliminate the effects of delays, and introduce negative feedback during communication, respectively. By employing fractional calculus tools, inequality techniques, and the proposed control law, sufficient criteria for the synchronization of the considered systems are rigorously established. Compared with existing synchronization works, this paper has significant advantages in model generality and controller design. Additionally, an explicit settling-time estimate is derived, which depends solely on control parameters and is independent of the initial conditions. Full article

Developing advanced functional materials requires the synergistic integration of nanoscale reinforcements with tailored properties. In this work, composite films of cellulose nanocrystals (CNCs), cellulose nanofibrils (CNFs), and reduced graphene oxide (rGO) were synthesized using a combination of solution casting, high shear homogenization, vacuum filtration, and environmentally friendly chemical reduction. The resulting CNC/CNF/rGO films exhibited a robust hierarchical structure with strong interfacial interactions, enabling exceptional mechanical properties, specifically a tensile strength of 215 MPa and a Young’s modulus of 18 GPa, alongside a continuous conductive network confirmed by frequency-independent electrical conductivity up to 30 kHz. Comprehensive dielectric characterization revealed frequency-dependent permittivity and low dielectric loss, aligning with Maxwell–Wagner theoretical predictions for heterogeneous composites. The composites also demonstrated thermal stability, with electrical conductivity increasing monotonically from 0 °C to 200 °C. These findings highlighted the CNC/CNF/rGO films’ suitability for applications in flexible electronics, electromagnetic shielding, packaging, and high-performance structural materials. Future optimization and modeling approaches, including fractional calculus, are recommended to further enhance multifunctionality and exploit the unique synergistic interactions intrinsic to nanocellulose–graphene oxide platforms. Full article

Numerical simulations of (1) two aerosol types such as organic carbon (i.e., spherical) and dust (i.e., non-spherical) particles, and (2) their mixtures are carried out. Optical and microphysical parameters of these aerosols in our simulations are provided by MERRA-2 (Modern-Era Retrospective Analysis for Research and Applications, version 2). The inversion routine is performed with TiARA (Tikhonov Advanced Regularization Algorithm) using the Lorenz–Mie (i.e., spherical) light-scattering model in unsupervised and automated, i.e., autonomous mode. The results of our numerical simulations show that the accuracy of the inversion results for the aerosol mixtures from synthetic optical data perturbed by ±10% random error is comparable to the accuracy observed for the inversion results of the “pure” spherical particles. In particular, the retrieval uncertainties of effective radius, and number, surface-area, and volume concentrations of these mixtures are ±30%, ±10%, between −50% and +100% and ±30%, respectively. However, we need to apply a modified version of the gradient correlation method (GCM) to stabilize the inversion results. The results of this study will form the baseline for future work, where we plan to apply TiARA to optical data products obtained from real lidar observations in the framework of the SCC (Single Calculus Chain) of EARLINET (European Aerosol Research Lidar Network). Full article

This paper presents two enhanced variants of the Artificial Protozoa Optimizer (APO), namely the Adaptive Balanced Artificial Protozoa Optimizer (AB-APO) and the Fractional Calculus-Enhanced Artificial Protozoa Optimizer (FC-APO), for optimal multi-Distributed Energy Resources (DERs) planning in smart radial distribution networks. The proposed framework addresses the coordinated allocation of Electric Vehicle Charging Stations (EVCSs), photovoltaic (PV) units, and Battery Energy Storage Systems (BESS). The AB-APO introduces an adaptive balancing mechanism that dynamically regulates exploration and exploitation to improve convergence stability and robustness, while the FC-APO incorporates fractional-order dynamics to embed long-memory effects, enhancing numerical stability and search smoothness. The proposed optimizers are evaluated on the IEEE-33 and IEEE-69 bus systems under eight DERs penetration scenarios. Simulation results demonstrate significant reductions in real and reactive power losses, improved voltage profiles, and effective mitigation of EV-induced network stress. Real power loss reductions exceeding 54%, 38.53%, 53.78%, 38.20%, 61.68%, and 60.72% are achieved for the IEEE-33 system, while reductions of 64.32%, 63.51%, 64.33%, 63.51%, 67.31%, and 67.04% are obtained for the IEEE-69 system across Scenarios 3–8. Overall, the results highlight the effectiveness of adaptive balancing and fractional-order modeling in strengthening APO-based optimization and confirm the suitability of the AB-APO and FC-APO as efficient planning tools for future smart distribution networks. Full article

This paper develops a fractional six-compartment model to describe malware spread in wireless sensor networks. To represent actual network activity, the model is constructed using generalized proportional-Caputo operators that incorporate memory and tempering effects. The existence and uniqueness of solutions are proved by applying fixed-point theorems. The stability of the system is then studied using the Ulam–Hyers approach and its extended form. A fractional Adams predictor–corrector method is employed to illustrate the dynamics. The results suggest that memory and tempering play an important role in shaping infection patterns, and they indicate that fractional calculus can provide a useful framework for studying and managing malware in distributed sensor networks. Full article

Quaternion-valued neural networks (QVNNs) that have multiple types of delays (leakage, time-varying, distributed, and neutral) and defined on time scales are discussed in this paper. Quaternions form a 4D normed division algebra and allow for a better representation of 3D and 4D data. QVNNs have been proposed and applications have appeared lately. Time-scale calculus was developed to allow the joint treatment of systems, or any hybrid mixing of them, and was also applied with success to the analysis of dynamic properties for neural networks (NNs). Because of its generality, encompassing the common properties of discrete-time (DT) and continuous-time (CT) NNs, time-scale NNs dynamics research does not benefit from a fully-developed Lyapunov theory. So, Halanay-type inequalities have to be used instead. To this end, we provide a novel generalization of inequalities of Halanay-type on time scales specifically suited for neutral systems, i.e., systems with neutral delays. Then, this new lemma is employed to obtain sufficient conditions presented both as linear matrix inequalities (LMIs) and as algebraic inequalities for the exponential stability and exponential synchronization of QVNNs on time scales with the mentioned delay types. The model put forward in this paper has a generality which is appealing for practical applications, in which both DT and CT dynamics are interesting, and all the discussed types of delays appear. For both the DT and CT scenarios, four numerical applications are used to illustrate the four theorems put forward in this research. Full article

Modern power distribution networks are increasingly challenged with nonlinear operating conditions, the high penetration of distributed energy resources, and conflicting operational objectives such as loss minimization and voltage regulation. Existing load-flow optimization approaches often suffer from slow convergence, premature stagnation in non-convex search spaces, and limited robustness when handling conflicting multi-objective performance criteria under fixed network constraints. To address these challenges, this paper proposes a Fractional Multi-Objective Load Flow Optimizer (FMOLFO), which integrates a fractional-order numerical regularization mechanism with an adaptive Pareto-based Differential Evolution framework. The fractional-order formulation employed in FMOLFO operates over an auxiliary iteration domain and serves as a numerical regularization strategy to improve the sensitivity conditioning and convergence stability of the load-flow solution, rather than modeling the physical time dynamics or memory effects of the power system. The optimization framework simultaneously minimizes physically consistent active power loss and voltage deviation within existing network operating constraints. Extensive simulations on IEEE 33-bus and 69-bus benchmark distribution systems demonstrate that FMOLFO achieves an up to 27% reduction in active power loss, improved voltage profile uniformity, and faster convergence compared with classical Newton–Raphson and metaheuristic baselines evaluated under identical conditions. The proposed framework is intended as a numerically enhanced, optimization-driven load-flow analysis tool, rather than a control- or dispatch-oriented optimal power flow formulation. Full article

For the first time, this paper introduces Nabla fractional calculus into the distributed Nash equilibrium (NE) seeking problem of aggregative games (AGs) with partial decision information in undirected communication networks, and proposes two novel fractional-order distributed algorithms. In the considered setting, each agent can access to only local information and collaboratively estimates the global aggregate through communication with its neighbors. Both algorithms adopt a backward-difference scheme followed by an implicit fractional-order gradient descent step. One updates local aggregate estimates via fractional-order dynamic tracking and the other uses fractional-order average dynamic consensus protocols. Under standard assumptions, convergence of both algorithms to the NE is rigorously proved using nabla fractional-order Lyapunov stability theory, achieving a Mittag-Leffler convergence rate. The feasibility of the developed schemes is verified via numerical experiments applied to a Nash-Cournot game and the coordination control of flexible robotic arms. Full article

Dynamic edge-server allocation in mobile edge computing (MEC) networks is a challenging multi-objective optimization problem due to highly dynamic user demands, spatiotemporal traffic variations, and the need to simultaneously minimize service latency and workload imbalance. Existing heuristic and metaheuristic-based approaches for this problem often suffer from premature convergence, limited exploration–exploitation balance, and inadequate adaptability to dynamic network conditions, leading to suboptimal edge-server placement and inefficient resource utilization. Moreover, most existing methods lack memory-aware search mechanisms, which restrict their ability to capture long-term system dynamics. To address these limitations, this paper proposes a Fractional-Order Multi-Objective African Vulture Optimization Algorithm (FO-MO-AVOA) for dynamic edge-server allocation. By integrating fractional-order calculus into the standard multi-objective AVOA framework, the proposed method introduces long-memory effects that enhance convergence stability, search diversity, and adaptability to time-varying workloads. The performance of FO-MO-AVOA is evaluated using realistic MEC network scenarios and benchmarked against several well-established metaheuristic algorithms. Simulation outcomes reveal that FO-MO-AVOA achieves 40–46% lower latency, 38–45% reduction in workload imbalance, and up to 28–35% reduction in maximum workload compared to competing methods. Extensive experiments conducted on real-world telecom network data demonstrate that FO-MO-AVOA consistently outperforms state-of-the-art multi-objective optimization algorithms in terms of convergence behaviour, Pareto-front quality, and overall system performance. Full article

The incorporation of fractional-order derivatives into neural networks presents a novel approach to improving gradient flow and adaptive learning dynamics. This paper introduces a fractional-order LSTM model, leveraging the Grünwald–Letnikov (GL) method to modify both activation functions and backpropagation mechanics. By redefining the transition functions of LSTM gates with fractional derivatives, the model achieves a smoother gradient adaptation while maintaining consistency across forward and backward passes. This is the first study integrating the Grünwald–Letnikov operator directly into both forward and backward LSTM computations, ensuring a consistent fractional framework throughout the entire learning process. We apply this approach to anomaly detection in fiber optic cable manufacturing, where small deviations in production parameters can significantly impact quality. A dataset containing time-series sensor measurements was used to train the fractional LSTM, demonstrating improved generalization and stability compared to classical LSTM models. Numerical stability analysis confirms that the fractional derivative framework allows convergent learning, preventing both vanishing and exploding gradients. Experimental results show that the fractional-order LSTM outperforms standard architectures in detecting manufacturing anomalies, with the optimal fractional order $\nu =0.95$ providing a balance between accuracy and computational complexity. The findings suggest that fractional calculus can enhance deep learning architectures by introducing a continuous and flexible transition between neuron activations, paving the way for adaptive neural networks with tunable memory effects. Full article

This paper pioneers the introduction of nabla fractional calculus into distributed Nash equilibrium (NE) seeking for non-cooperative games (NGs), proposing several novel discrete-time fractional-order algorithms. We first develop a gradient play-based algorithm under perfect information and subsequently extend it to partial-information settings. Two types of communication network topologies among agents, namely connected undirected graphs and strongly connected unbalanced directed graphs, are explicitly considered. When the pseudo-gradient mapping of the NG is Lipschitz continuous and strongly monotone, the proposed algorithms are proven to achieve asymptotic convergence to the NE with at least a Mittag–Leffler convergence rate. Both the step size and the fractional order act as tunable parameters that jointly influence the convergence performance. Numerical experiments on potential games and Nash–Cournot games demonstrate the effectiveness of the proposed algorithms. Full article