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

• Published in: Mathematical methods in the applied sciences Vol. 43; no. 3; pp. 1437 - 1466
• Main Authors: Hadian Rasanan, Amir Hosein, Bajalan, Nastaran, Parand, Kourosh, Rad, Jamal Amani
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.02.2020
• Subjects: Algorithms; Caputo derivative; cognitive decision making; Computer simulation; Decision making; Finite element method; fractional calculus; fractional dynamics; fractional neural network; fractional order of Jacobi functions; Levenberg‐Marquardt algorithm; Mathematical models; Meshless methods; Neural networks; Nonlinear dynamics; Nonlinear systems; Numerical methods; Partial differential equations; Polynomials
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.5981

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.