3,689 research outputs found
Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics
A two-state master equation based decision making model has been shown to
generate phase transitions, to be topologically complex and to manifest
temporal complexity through an inverse power-law probability distribution
function in the switching times between the two critical states of consensus.
These properties are entailed by the fundamental assumption that the network
elements in the decision making model imperfectly imitate one another. The
process of subordination establishes that a single network element can be
described by a fractional master equation whose analytic solution yields the
observed inverse power-law probability distribution obtained by numerical
integration of the two-state master equation to a high degree of accuracy
Foundations of Fractional Calculus on Time Scales
Bakalářská práce pojednává o zlomkovém kalkule na časových škálach, přesněji - zavádí zlomkový kalkulus na časových škálach a taktéž vyšetřuje jednoznačnost axiomatické definice zavádějící mocninné funkce. Po zavedení základních pojmů je předmětem diskuze hlavně zobecněná Laplaceova transformace a důkaz jednoznačnosti zobecněné Laplaceovy transformace, která je použita jako nástroj pro dokázání jednoznačnosti zlomkových mocniných funkcií na časových škálach.The bachelor thesis concerns fractional calculus on time scales, more precisely, it introduces fractional calculus on time scales and also investigates the property of uniqueness of the axiomatic definition of the power functions. After introducing basic concepts, the subject of discussion is mostly generalized Laplace transform as well as proof of uniqueness of generalized Laplace transform, which is used as a tool to proving the uniqueness of fractional power functions on time scales.
Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study
In this work, we rigorously investigate the robustness of graph neural
fractional-order differential equation (FDE) models. This framework extends
beyond traditional graph neural (integer-order) ordinary differential equation
(ODE) models by implementing the time-fractional Caputo derivative. Utilizing
fractional calculus allows our model to consider long-term memory during the
feature updating process, diverging from the memoryless Markovian updates seen
in traditional graph neural ODE models. The superiority of graph neural FDE
models over graph neural ODE models has been established in environments free
from attacks or perturbations. While traditional graph neural ODE models have
been verified to possess a degree of stability and resilience in the presence
of adversarial attacks in existing literature, the robustness of graph neural
FDE models, especially under adversarial conditions, remains largely
unexplored. This paper undertakes a detailed assessment of the robustness of
graph neural FDE models. We establish a theoretical foundation outlining the
robustness characteristics of graph neural FDE models, highlighting that they
maintain more stringent output perturbation bounds in the face of input and
graph topology disturbances, compared to their integer-order counterparts. Our
empirical evaluations further confirm the enhanced robustness of graph neural
FDE models, highlighting their potential in adversarially robust applications.Comment: in Proc. AAAI Conference on Artificial Intelligence, Vancouver,
Canada, Feb. 202
Recent Advances and Applications of Fractional-Order Neural Networks
This paper focuses on the growth, development, and future of various forms of fractional-order neural networks. Multiple advances in structure, learning algorithms, and methods have been critically investigated and summarized. This also includes the recent trends in the dynamics of various fractional-order neural networks. The multiple forms of fractional-order neural networks considered in this study are Hopfield, cellular, memristive, complex, and quaternion-valued based networks. Further, the application of fractional-order neural networks in various computational fields such as system identification, control, optimization, and stability have been critically analyzed and discussed
Enhanced Fractional Adaptive Processing Paradigm for Power Signal Estimation
Fractional calculus tools have been exploited to effectively model variety of engineering, physics and applied sciences problems. The concept of fractional derivative has been incorporated in the optimization process of least mean square (LMS) iterative adaptive method. This study exploits the recently introduced enhanced fractional derivative based LMS (EFDLMS) for parameter estimation of power signal formed by the combination of different sinusoids. The EFDLMS addresses the issue of fractional extreme points and provides faster convergence speed. The performance of EFDLMS is evaluated in detail by taking different levels of noise in the composite sinusoidal signal as well as considering various fractional orders in the EFDLMS. Simulation results reveal that the EDFLMS is faster in convergence speed than the conventional LMS (i.e., EFDLMS for unity fractional order)
Dynamical Systems
Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...
FSCSCOOT: Functional Calculus Competitive Swarm Coot Optimization-based CNN transfer learning for Parkinson’s disease classification
Parkinson's disease (PD) is a neurological disorder of the central nervous system that causes difficulty in movement, often including tremors and rigidity. Early detection of PD can prevent symptoms up to a certain age and increase life expectancy. For this purpose, we have used brain images from magnetic resonance imaging (MRI) technique. Generally dementia can be either classified as Alzheimer’s or Parkinson’s or sometimes may be due to tumor in brain. Therefore, effectual methods such as Competitive Swarm Coot Optimization_ Convolutional Neural Network (CSCOOT_CNN) with transfer learning and Fractional CSCOOT_ deep neuro-fuzzy network (FCSCOOT_DNFN are newly introduced for classification of brain diseases. At first, input images are acquired from particular datasets, and then input images are given to the pre-processing stage. In a pre-processing module, median filter is utilized for the elimination of noises. Afterward, pre-processed image is then subjected to feature extraction in which CNN features are extracted. In the level of classification, the images are classified into Parkinson by DNFN that is trained utilizing the introduced FCSCOOT algorithm. Furthermore, the FCSCOOT algorithm is newly designed by combination of Fractional Calculus (FC) with CSCOOT algorithm
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