Two methods for studying the response and the reliability of a fractional stochastic dynamical system

In this paper, we use stochastic averaging method and data-driven artificial neural network to study the response and reliability of a fractional stochastic dynamical system. Firstly, a generalized Harmonic transformation is introduced to get an approximate expression for the fractional derivative. Based on this procedure, stochastic averaging method is utilized to get Fokker-Planck Kolmogorov equation for the system response and Backward Kolmogorov equation for the system reliability. An exact stationary solu-tion of amplitude response is obtained by solving Fokker-Planck Kolmogorov equation, in the meanwhile, numerical results regarding to the reliability by Crank-Nicolson difference method are derived too. We propose an algorithm of data-driven artificial neural network and apply it for solving these two equations simulatively, in order to verify the correction and accuracy of the analytical methods. All results illustrate that data-driven artificial neural network is highly effective to achieve the system response and reliability. The advantage of this deep learning algorithm mainly includes the meshless structure, boundary sample data collection, small amount of training data and unconstrained optimization. In addition, the increase of the fractional order can enhance the system response and leads to stochastic bifurcation.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

In this paper, the response and reliability of a fractional stochastic dynamical system are studied using the stochastic averaging method and data-driven artificial neural network. A generalized Harmonic transformation is introduced to approximate the fractional derivative, and the Fokker-Planck Kolmogorov equation for the system response and the Backward Kolmogorov equation for the system reliability are obtained using the stochastic averaging method. An exact stationary solution for the amplitude response is obtained by solving the Fokker-Planck Kolmogorov equation, and numerical results for the reliability are derived using the Crank-Nicolson difference method. A data-driven artificial neural network algorithm is proposed and applied to simulate the solution of these two equations, in order to verify the accuracy of the analytical methods. The results demonstrate the high effectiveness of the data-driven artificial neural network in achieving the system response and reliability. The advantages of this deep learning algorithm include its meshless structure, boundary sample data collection, small amount of training data, and unconstrained optimization. Furthermore, increasing the fractional order enhances the system response and leads to stochastic bifurcation.