Global asymptotic stability of impulsive fractional-order complex-valued neural networks with time delay

Fractional-Order Modeling of Heat and Moisture Transfer in Anisotropic Materials Using a Physics-Informed Neural Network

Mathematical models of heat and moisture transfer for anisotropic materials, based on the use of the fractional calculus of integro-differentiation, are considered because such two-factor fractal models have not been proposed in the literature so far. The numerical implementation of mathematical models for determining changes in heat exchange and moisture exchange is based on the adaptation of the fractal neural network method, grounded in the physics of processes. A fractal physics-informed neural network architecture with a decoupled structure is proposed, based on loss functions informed by the physical process under study. Fractional differential formulas are applied to the expressions of non-integer operators, and finite difference schemes are developed for all components of the loss functions. A step-by-step method for network training is proposed. An algorithm for the implementation of the fractal physics-informed neural network is developed. The efficiency of the new method is substantiated by comparing the obtained numerical results with numerical approximation by finite differences and experimental data for particular cases.

On uniform stability and numerical simulations of complex valued neural networks involving generalized Caputo fractional order

The dynamics and existence results of generalized Caputo fractional derivatives have been studied by several authors. Uniform stability and equilibrium in fractional-order neural networks with generalized Caputo derivatives in real-valued settings, however, have not been extensively studied. In contrast to earlier studies, we first investigate the uniform stability and equilibrium results for complex-valued neural networks within the framework of a generalized Caputo fractional derivative. We investigate the intermittent behavior of complex-valued neural networks in generalized Caputo fractional-order contexts. Numerical results are supplied to demonstrate the viability and accuracy of the presented results. At the end of the article, a few open questions are posed.

Novel results on asymptotic stability and synchronization of fractional-order memristive neural networks with time delays: The 0<δ≤1 case

This paper investigates the asymptotic stability and synchronization of fractional-order (FO) memristive neural networks with time delays. Based on the FO comparison principle and inverse Laplace transform method, the novel sufficient conditions for the asymptotic stability of a FO nonlinear system are given. Then, based on the above conclusions, the sufficient conditions for the asymptotic stability and synchronization of FO memristive neural networks with time delays are investigated. The results in this paper have a wider coverage of situations and are more practical than the previous related results. Finally, the validity of the results is checked by two examples.

Synchronization in Finite-Time of Delayed Fractional-Order Fully Complex-Valued Dynamical Networks via Non-Separation Method

The finite-time synchronization (FNTS) problem for a class of delayed fractional-order fully complex-valued dynamic networks (FFCDNs) with internal delay and non-delayed and delayed couplings is studied by directly constructing Lyapunov functions instead of decomposing the original complex-valued networks into two real-valued networks. Firstly, a mixed delay fractional-order mathematical model is established for the first time as fully complex-valued, where the outer coupling matrices of the model are not restricted to be identical, symmetric, or irreducible. Secondly, to overcome the limitation of the use range of a single controller, two delay-dependent controllers are designed based on the complex-valued quadratic norm and the norm composed of its real and imaginary parts' absolute values, respectively, to improve the synchronization control efficiency. Besides, the relationships between the fractional order of the system, the fractional-order power law, and the settling time (ST) are analyzed. Finally, the feasibility and effectiveness of the control method designed in this paper are verified by numerical simulation.

Exponentially Stable Periodic Oscillation and Mittag-Leffler Stabilization for Fractional-Order Impulsive Control Neural Networks With Piecewise Caputo Derivatives

It is well known that the conventional fractional-order neural networks (FONNs) cannot generate nonconstant periodic oscillation. For this point, this article discusses a class of impulsive FONNs with piecewise Caputo derivatives (IPFONNs). By using the differential inclusion theory, the existence of the Filippov solutions for a discontinuous IPFONNs is investigated. Furthermore, some decision theorems are established for the existence and uniqueness of the (periodic) solution, global exponential stability, and impulsive control global stabilization to IPFONNs. This article achieves four key issues that were not solved in the previously existing literature: 1) the existence of at least one Filippov solution in a discontinuous IPFONN; 2) the existence and uniqueness of periodic oscillation in a nonautonomous IPFONN; 3) global exponential stability of IPFONNs; and 4) impulsive control global Mittag-Leffler stabilization for FONNs.

Artificial neural networks: a practical review of applications involving fractional calculus

In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order.

Finite-Time Mittag–Leffler Synchronization of Neutral-Type Fractional-Order Neural Networks with Leakage Delay and Time-Varying Delays

This paper studies fractional-order neural networks with neutral-type delay, leakage delay, and time-varying delays. A sufficient condition which ensures the finite-time synchronization of these networks based on a state feedback control scheme is deduced using the generalized Gronwall–Bellman inequality. Then, a different state feedback control scheme is employed to realize the finite-time Mittag–Leffler synchronization of these networks by using the fractional-order extension of the Lyapunov direct method for Mittag–Leffler stability. Two numerical examples illustrate the feasibility and the effectiveness of the deduced sufficient criteria.

Multistability Analysis of Quaternion-Valued Neural Networks With Time Delays

This paper addresses the multistability issue for quaternion-valued neural networks (QVNNs) with time delays. By using the inequality technique, sufficient conditions are proposed for the boundedness and the global attractivity of delayed QVNNs. Based on the geometrical properties of the activation functions, several criteria are obtained to ensure the existence of equilibrium points, of which are locally stable. Two numerical examples are provided to illustrate the effectiveness of the obtained results.

A novel and reliable computational intelligence system for breast cancer detection

Cancer is the second important morbidity and mortality factor among women and the most incident type is breast cancer. This paper suggests a hybrid computational intelligence model based on unsupervised and supervised learning techniques, i.e., self-organizing map (SOM) and complex-valued neural network (CVNN), for reliable detection of breast cancer. The dataset used in this paper consists of 822 patients with five features (patient's breast mass shape, margin, density, patient's age, and Breast Imaging Reporting and Data System assessment). The proposed model was used for the first time and can be categorized in two stages. In the first stage, considering the input features, SOM technique was used to cluster the patients with the most similarity. Then, in the second stage, for each cluster, the patient's features were applied to complex-valued neural network and dealt with to classify breast cancer severity (benign or malign). The obtained results corresponding to each patient were compared to the medical diagnosis results using receiver operating characteristic analyses and confusion matrix. In the testing phase, health and disease detection ratios were 94 and 95%, respectively. Accordingly, the superiority of the proposed model was proved and can be used for reliable and robust detection of breast cancer.