Search Results (6)

Due to the widespread application of neural networks (NNs), and considering the respective advantages of fractional calculus (FC), inertial neural networks (INNs), cellular neural networks (CNNs), and fuzzy neural networks (FNNs), this paper investigates the fixed-time synchronization (FDTS) issues for a particular category of fractional-order cellular-inertial fuzzy neural networks (FCIFNNs) that involve mixed time-varying delays (MTDs), including both discrete and distributed delays. Firstly, we establish an appropriate transformation variable to reformulate FCIFNNs with MTD into a differential first-order system. Then, utilizing the finite-time stability (FETS) theory and Lyapunov functionals (LFs), we establish some new effective criteria for achieving FDTS of the response system (RS) and drive system (DS). Eventually, we offer two numerical examples to display the effectiveness of our proposed synchronization strategies. Moreover, we also demonstrate the benefits of our approach through an application in image encryption. Full article

The physics governing the fluid dynamics of bio-inspired flapping wings is effectively characterized by partial differential equations (PDEs). Nevertheless, the process of discretizing these equations at spatiotemporal scales is notably time consuming and resource intensive. Traditional PDE-based computations are constrained in their applicability, which is mainly due to the presence of numerous shape parameters and intricate flow patterns associated with bionic flapping wings. Consequently, there is a significant demand for a rapid and accurate solution to nonlinear PDEs, to facilitate the analysis of bionic flapping structures. Deep learning, especially physics-informed deep learning (PINN), offers an alternative due to its great nonlinear curve-fitting capability. In the present work, a hybrid coarse-data-driven physics-informed neural network model (HCDD-PINN) is proposed to improve the accuracy and reliability of predicting the time evolution of nonlinear PDEs solutions, by using an order-of-magnitude-coarser grid than traditional computational fluid dynamics (CFDs) require as internal training data. The architecture is devised to enforce the initial and boundary conditions, and incorporate the governing equations and the low-resolution spatiotemporal internal data into the loss function of the neural network, to drive the training. Compared to the original PINN with no internal data, the training and predicting dynamics of HCDD-PINN with different resolutions of coarse internal data are analyzed on the problem relevant to the two-dimensional unsteady flapping wing, which involves unsteady flow features and moving boundaries. Additionally, a hyper-parametrical study is conducted to obtain an optimal model for the problem under consideration, which is then utilized for investigating the effects of the snapshot and fraction of the coarse internal data on the HCDD-PINN’s performances. The results show that the proposed framework has a sufficient stability and accuracy for solving the considered biomimetic flapping-wing problem, and its great potential means that it can be considered as an alternative to accelerate or replace traditional CFD solvers in the future. The interested variables of the flow field at any instant can be rapidly obtained by the trained HCDD-PINN model, which is superior to the traditional CFD method that usually needs to be re-run. For the three-dimensional and optimization problems of flapping wings, the advantages of the proposed method are supposedly even more apparent. Full article

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. This research makes a contribution to the topic by describing and establishing the first generalized discrete fractional variable order Gronwall inequality that we employ to examine the finite time stability of nonlinear Nabla fractional variable-order discrete neural networks. This is followed by a specific version of a generalized variable-order fractional discrete Gronwall inequality described using discrete Mittag–Leffler functions. A specific version of a generalized variable-order fractional discrete Gronwall inequality represented using discrete Mittag–Leffler functions is shown. As an application, utilizing the contracting mapping principle and inequality approaches, sufficient conditions are developed to assure the existence, uniqueness, and finite-time stability of the equilibrium point of the suggested neural networks. Numerical examples, as well as simulations, are provided to show how the key findings can be applied. Full article

Given the recent advances regarding the studies of discrete fractional calculus, and the fact that the dynamics of discrete-time neural networks in fractional variable-order cases have not been sufficiently documented, herein, we consider a novel class of discrete-time fractional-order neural networks using discrete nabla operator of variable-order. An adequate criterion for the existence of the solution in addition to its uniqueness for such systems is provided with the use of Banach fixed point technique. Moreover, the uniform stability is investigated. We provide at the end two numerical simulations illustrating the relevance of the aforementioned results. Full article

In this work, we recall some definitions on fractional calculus with discrete-time. Then, we introduce a discrete-time Hopfield neural network (D.T.H.N.N) with non-commensurate fractional variable-order (V.O) for three neurons. After that, phase-plot portraits, bifurcation and Lyapunov exponents diagrams are employed to verify that the proposed discrete time Hopfield neural network with non-commensurate fractional variable order has chaotic behavior. Furthermore, we use the 0-1 test and $C_0$ complexity algorithm to confirm and prove the results obtained about the presence of chaos. Finally, simulations are carried out in Matlab to illustrate the results. Full article

Few papers have been published to date regarding the stability of neural networks described by fractional difference operators. This paper makes a contribution to the topic by presenting a variable-order fractional discrete neural network model and by proving its Ulam–Hyers stability. In particular, two novel theorems are illustrated, one regarding the existence of the solution for the proposed variable-order network and the other regarding its Ulam–Hyers stability. Finally, numerical simulations of three-dimensional and two-dimensional variable-order fractional neural networks were carried out to highlight the effectiveness of the conceived theoretical approach. Full article