Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Abstract

By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.

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

“…In Ref. [22], an orthogonal Jacobi FANN was employed to perform the numerical simulations of nonlinear frac-tional dynamics based on various types of FDE, the obtained results were compared with other numerical methods, such as the spectral collocation method, meshless method, and reproducing kernel method, to demonstrate the feasibility of the proposed approach.…”

“…Finally, Hadian Rasanan in Ref. [22] implemented a fractional ANN. The authors used fractional-order Jacobi functions as the activation function of the hidden layer.…”

Novel ANN Method for Solving Ordinary and Time‐Fractional Black–Scholes Equation

“…where α denotes the Caputo derivatives of the function and 0 < α < 1. As the first step for such FPDEs, the Caputo derivative should be discretized (readers are advised to see Hadian Rasanan et al [48] for the preliminaries and thorough information about this type of derivative). Consider the following theorem.…”

Optimal Control Problems Involving Combined Fractional Operators with General Analytic Kernels

“…In this direction, Fernandez, Özarslan and Baleanu proposed in 2019 a fractional integral operator, based on a general analytic kernel, that includes a number of existing and known operators [1]. Since this seminal work of 2019, several interesting results appeared, e.g., determination of source terms for fractional Rayleigh-Stokes equations with random data [2], new analytic properties of tempered fractional calculus [3], simulation of nonlinear dynamics with fractional neural networks arising in the modeling of cognitive decision making processes [4], new numerical methods for variable order fractional nonlinear quadratic integro-differential equations [5], and analysis of impulsive ϕ-Hilfer fractional differential equations [6]. Here, we investigate, for the first time in the literature, optimal control problems that involve a combined Caputo fractional derivative with a general analytic kernel in the sense of Fernandez, Özarslan and Baleanu.…”