Fractional-order Neural Networks Research Articles

Effect of Leakage Delays on Bifurcation in Fractional-Order Bidirectional Associative Memory Neural Networks with Five Neurons and Discrete Delays

Effect of Leakage Delays on Bifurcation in Fractional-Order Bidirectional Associative Memory Neural Networks with Five Neurons and Discrete Delays

Finite-time synchronization of fractional order neural networks via sampled data control with time delay

Finite-time synchronization of fractional order neural networks via sampled data control with time delay

Stability analysis of discrete-time tempered fractional-order neural networks with time delays

Stability analysis of discrete-time tempered fractional-order neural networks with time delays

Extended dissipative performance of fractional-order neural networks via LMI approach

Extended dissipative performance of fractional-order neural networks via LMI approach

Power-exponential and fixed-time consensus of conformable fractional-order quantum cellular neural networks via event-triggered control

Quantum neural networks (QNNs) are considered to be superior to classical ANNs in machine learning, memory capacity, information processing, and quantum system simulation. However, In a practical and complex system, the dynamic behavior of an open quantum system could not be accurately described by an integer-ordered Schrödinger equation. In this paper, the conformable time-fractional-order Schrödinger equation is proposed, and accordingly, the model of conformable fractional-order quantum cellular neural networks (CFOQCNNs) is established and derived from the as-proposed equation. The properties of the conformable fractional-order derivative are studied and several new inequalities regarding the power-exponential and fixed-time convergence of conformable fractional-order systems are obtained. To save the communication resource, we introduce the event-triggered mechanism to construct the controllers and then the power-exponential and fixed-time synchronizations of the master-slave systems derived from the above CFOQCNNs are studied. We also prove the absence of Zeno behaviors regarding the event-triggered strategies. According to the numerical simulation, the dynamic behavior of the CFOQCNNs is discussed and the dissipativity of the CFOQCNNs is briefly revealed. Then the synchronization behaviors of the master and slave CFOQCNNs under power-exponential and fixed-time event-triggered control are demonstrated, where the effectiveness of the event-triggered control strategy is verified. Control behaviors with different fractional orders are also presented. We also discuss the hybrid of power-exponential control and fixed-time control and illustrate the advantages of the hybrid strategy. In the last, we conclude our studies, analyze the drawbacks of this work, and briefly introduce our future research.

Comparative study of integer-order and fractional-order artificial neural networks: Application for mathematical function generation

This paper investigates the impact of fractional derivatives on the activation functions of an artificial neural network (ANN). Based on the results and analysis, a three-layer backpropagation neural network model with fractional and integer derivatives employed in the activation function and fractional gradient descent method backpropagation learning algorithm has been proposed. Specifically, three perceptrons have been proposed based on fractional and integer derivatives applied to activation functions and learning algorithms. They are fractional derivative activation function (FDAF) perceptron, integer derivative activation function (IDAF)-fractional derivative learning algorithm (FDLA) perceptron, and fractional derivative learning algorithm (FDLA) perceptron. The Riemann-Liouville (RL) fractional derivative, Grunwald-Letnikov (GL) derivative, Caputo-Fabrizio (CF) fractional derivative, Caputo (C) fractional derivative, and Atangana-Baleanu (AB) fractional derivative have been employed. The impact of these derivatives’ fractional-order (FO) is investigated in a wide range from 0.1−0.9 on testing mean square error (MSE) and time required to train the perceptron. FO-based and IO-based perceptrons are compared with the help of performance metrics such as testing MSE and the time required to train the models. The training and testing simulation results illustrate that Caputo-Fabrizio derivative-based perceptrons outperform other fractional derivatives. Also, the performance of the FO-based perceptron is better in terms of the least MSE.

Time-Varying Function Matrix Projection Synchronization of Caputo Fractional-Order Uncertain Memristive Neural Networks with Multiple Delays via Mixed Open Loop Feedback Control and Impulsive Control

This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization (TFMPS), which is a generalized version of traditional matrix projection synchronization, modified projection synchronization, complete synchronization, and anti-synchronization. To achieve the goal of TFMPS, we design a novel mixed controller including the open loop feedback control and impulsive control, which employs the state information in the time-delayed interval and the sampling information at the impulse instants. It has a prominent advantage that impulse intervals are not restricted by time delays. To establish the connection between the error system and the auxiliary system, a generalized fractional-order comparison theorem with time-varying coefficients and impulses is established. Applying the stability theory, the comparison theorem, and the Laplace transform, new synchronization criteria of FOUMNNs are acquired under the mixed impulsive control schemes, and the derived synchronization theorem and corollary can effectively expand the correlative synchronization achievements of fractional-order systems.

Convergence Analysis of Novel Fractional-Order Backpropagation Neural Networks With Regularization Terms.

Fractional-order derivatives have the potential to improve the performance of backpropagation (BP) neural networks. Several studies have found that the fractional-order gradient learning methods may not converge to real extreme points. The truncation and the modification of the fractional-order derivative are applied to guarantee convergence to the real extreme point. Nonetheless, the real convergence ability is based on the assumption that the algorithm is convergent, which limits the practicality of the algorithm. In this article, a novel truncated fractional-order BP neural network (TFO-BPNN) and a novel hybrid TFO-BPNN (HTFO-BPNN) are designed to solve the above problem. First, to avoid overfitting, a squared regularization term is introduced into the fractional-order BP neural network. Second, a novel dual cross-entropy cost function is proposed and employed as a loss function for the two neural networks. The penalty parameter helps to adjust the effect of the penalty term and further alleviates the gradient vanishing problem. In terms of convergence, the convergence ability of the two proposed neural networks is first proved. Then, the convergence ability to the real extreme point is further analyzed theoretically. Finally, the simulation results effectively illustrate the feasibility, high accuracy, and good generalization ability of the proposed neural networks. Comparative studies among the proposed neural networks and some related methods further substantiate the superiority of the TFO-BPNN and the HTFO-BPNN.

Synchronization of fractional-order quaternion-valued neural networks with image encryption via event-triggered impulsive control

The finite-time synchronization and impulsive synchronization of fractional-order quaternion-valued neural networks (FOQVNNs) with delay are investigated in this research. To begin, instead of employing a decomposition method, a Lyapunov direct approach is proposed to achieve the finite-time synchronization of FOQVNNs using event-triggered control based on state errors. The settling time is then estimated using a unique fractional differential inequality. The event-triggered impulsive controller, which updates only during triggering instants, is proposed to be more inexpensive and efficient. Then, utilizing fractional Lyapunov theory and the comparison principle, several novel quaternion-valued linear matrix inequalities criteria are defined to assure the impulsive synchronization of the delayed FOQVNNs. Furthermore, the positive lower bound of the inter-event time is obtained to exclude the Zeno behavior. As proof of the feasibility and validity of these theoretical findings obtained, numerical examples are provided. At last, the use of the event-triggered control technique for FOQVNNs in image encryption and decryption is demonstrated and validated.

Asymptotic Stability Analysis and Stabilization Control for General Fractional-Order Neural Networks via an Unified Lyapunov Function

Asymptotic Stability Analysis and Stabilization Control for General Fractional-Order Neural Networks via an Unified Lyapunov Function

Quasi-Projective Synchronization of Discrete-Time Fractional-Order Complex-Valued BAM Fuzzy Neural Networks via Quantized Control

In this paper, we ponder a kind of discrete-time fractional-order complex-valued fuzzy BAM neural network. Firstly, in order to guarantee the quasi-projective synchronization of the considered networks, an original quantitative control strategy is designed. Next, by virtue of the relevant definitions and properties of the Mittag-Leffler function, we propose a novel discrete-time fractional-order Halanay inequality, which is more efficient for disposing of the discrete-time fractional-order models with time delays. Then, based on the new lemma, fractional-order h-difference theory, and comparison principle, we obtain some easy-to-verify synchronization criteria in terms of algebraic inequalities. Finally, numerical simulations are provided to check the accuracy of the proposed theoretical results.

Coexistence and locally exponential stability of multiple equilibrium points for fractional-order impulsive control Cohen–Grossberg neural networks

Different from the existing multiple Mittag-Leffler stability or multiple asymptotic stability, the multiple exponential stability, which has explicit and faster convergence rate, is investigated in this paper for fractional-order impulsive control Cohen–Grossberg neural networks. First, by using the definition of Dirac delta function, the fractional-order control Cohen–Grossberg neural networks are translated into fractional-order impulsive neural networks, in which pulse effect relies on the fractional order of the addressed system. Then, based on maximum norm, 1−norm and general q−norm (q=2n), a series of novel criteria are obtained respectively to ensure that such n−neuron neural networks can have ∏i=1n(2Li+1) total equilibrium points and ∏i=1n(Li+1) locally exponentially stable equilibrium points, by utilizing the known fixed point theorem, the method of average impulsive interval, the theory of fractional-order differential equations, and the method of Lyapunov function. This paper’s investigation reveals the effects of impulsive function, impulsive interval, and fractional order on the dynamic behaviors. Finally, theoretical results are shown to be effective by four illustrative examples.

Finite-Time Adaptive Synchronization and Fixed-Time Synchronization of Fractional-Order Memristive Cellular Neural Networks with Time-Varying Delays

Asymptotic synchronization requires continuous external control of the system, which is unrealistic considering the cost of control. Adaptive control methods have strong robustness to uncertainties such as disturbances and unknowns. On the other hand, for finite-time synchronization, if the initial value of the system is unknown, the synchronization time of the finite-time synchronization cannot be estimated. This paper explores the finite-time adaptive synchronization (FTAS) and fixed-time synchronization (FDTS) of fractional-order memristive cellular neural networks (FMCNNs) with time-varying delays (TVD). Utilizing the properties and principles of fractional order, we introduce a novel lemma. Based on this lemma and various analysis techniques, we establish new criteria to guarantee FTAS and FDTS of FMCNNs with TVD through the implementation of a delay-dependent feedback controller and fractional-order adaptive controller. Additionally, we estimate the upper bound of the synchronization setting time. Finally, numerical simulations are conducted to confirm the validity of the finite-time and fixed-time stability theorems.

Projective synchronization for distinct fractional-order neural networks consist of inconsistent orders via sliding mode control

The present study delves into the projective synchronization for distinct fractional-order neural networks consist of inconsistent orders, employing the principles of sliding mode control and incorporating fractional operators within the controller. Firstly, incorporating the fractional-order derivative into the controller facilitates the derivation of a synchronous error system, and a well-suited integral-type sliding switching surface is subsequently constructed, along with the development of an appropriate sliding mode controller, thereby ensuring the existence of sliding movement derived from the principle of sliding mode control. Then through the application of the Lyapunov direct method, it is shown that the sliding mode controller steers the evolution of error system towards the designated sliding switching surface and maintains therein indefinitely, meanwhile, novel criteria for achieving projective synchronization for distinct fractional-order neural networks consist of inconsistent orders is established. Lastly, to exemplify the efficacy and practical applications of the proposed approach, simulations with two types of distinct fractional-order neural networks are carried out utilizing a continuous frequency distribution model.

Finite-Time Synchronization of Fractional-Order Nonlinear Systems with State-Dependent Delayed Impulse Control

This paper delves into the topics of Finite-Time Stabilization (FTS) and Finite-Time Contractive Stabilization (FTCS) for Fractional-Order Nonlinear Systems (FONSs). To address these issues, we employ a State-Dependent Delayed Impulsive Controller (SDDIC). By leveraging both Lyapunov theory and impulsive control theory, we establish sufficient conditions for achieving stability criteria for fractional-order systems. Initially, we employ the aforementioned sufficient conditions to derive stability criteria for general FONSs within the SDDIC framework, employing Linear Matrix Inequality (LMI) techniques. Furthermore, we apply these theoretical findings to tackle the challenge of finite-time synchronization in fractional-order chaotic systems using the proposed SDDIC. We substantiate the efficacy of these theoretical advancements through numerical simulations that vividly demonstrate their capability to achieve finite-time synchronization in fractional-order cellular neural networks and fractional-order Chua’s circuits. Moreover, we introduce an innovative chaos-based multi-image encryption algorithm, thereby contributing significantly to the field. To ensure the algorithm’s robustness, we subject it to rigorous statistical tests, which confidently affirm its capacity to provide the requisite level of security.

Hybrid LSTM-Based Fractional-Order Neural Network for Jeju Island’s Wind Farm Power Forecasting

Efficient integration of wind energy requires accurate wind power forecasting. This prediction is critical in optimising grid operation, energy trading, and effectively harnessing renewable resources. However, the wind’s complex and variable nature poses considerable challenges to achieving accurate forecasts. In this context, the accuracy of wind parameter forecasts, including wind speed and direction, is essential to enhancing the precision of wind power predictions. The presence of missing data in these parameters further complicates the forecasting process. These missing values could result from sensor malfunctions, communication issues, or other technical constraints. Addressing this issue is essential to ensuring the reliability of wind power predictions and the stability of the power grid. This paper proposes a long short-term memory (LSTM) model to forecast missing wind speed and direction data to tackle these issues. A fractional-order neural network (FONN) with a fractional arctan activation function is also developed to enhance generated wind power prediction. The predictive efficacy of the FONN model is demonstrated through two comprehensive case studies. In the first case, wind direction and forecast wind speed data are used, while in the second case, wind speed and forecast wind direction data are used for predicting power. The proposed hybrid neural network model improves wind power forecasting accuracy and addresses data gaps. The model’s performance is measured using mean errors and R2 values.

Stability and Hopf Bifurcation Analysis of A Fractional-Order BAM Neural Network with Two Delays Under Hybrid Control

In this paper, considering that fractional-order calculus can more accurately describe memory and genetic properties, we introduce fractional integral operators into neural networks and discuss the stability and Hopf bifurcation of a fractional-order bidirectional associate memory (BAM) neural network with two delays. In addition, the hybrid controller is proposed to achieve Hopf bifurcation control of the system. By taking two time delays as the bifurcation parameters and analyzing of the corresponding characteristic equation, stability switching curves of the controllable system for two delays are obtained. The direction of the characteristic root crossing the imaginary axis in stability switching curves is determined. Sufficient criteria are sequentially given to judge the local stability and the existence of Hopf bifurcation of a fractional-order BAM neural network system. The numerical simulation results show that the hybrid controller can effectively control Hopf bifurcation of a fractional-order BAM neural network system with two delays.

Stabilization Analysis of Delayed Fractional-Order Neural Networks via Multi-Constant-Delay Combination Method

Stabilization Analysis of Delayed Fractional-Order Neural Networks via Multi-Constant-Delay Combination Method

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.

Bipartite Synchronization of Fractional Order Multiple Memristor Coupled Delayed Neural Networks with Event Triggered Pinning Control

This paper investigates the leader and leaderless bipartite synchronization with the signed network utilizing the model of multiple memristor and coupled delayed neural network in an event-triggered pinning control. The usage of the descriptor method in fractional-order neural networks in case of a non-differentiable delay can be seen in this paper. Further, Lyapunov functional criteria, including Lur’e Postnikov Lyapunov functional, is established, and bipartite leader and leaderless synchronization are proved. The obtained numerical results can be seen as accurate to the theoretical results.