Reliability Problem of a Fractional Stochastic Dynamical System Based on Stochastic Averaging Method and Data-Driven Deep Learning Algorithm

Engineering structures with viscoelastic materials are generally modelled by a fractional-order system. The reliability problem of relative structural vibration under random excitations is always a hot issue in the field of the stochastic dynamical systems. Consider a generalized Van der Pol system with fractional derivatives excited by a white Gaussian noise. Firstly, a generalized harmonic transformation is used to get an approximated expression for fractional derivative by converting the fast-varying variables to the slow-varying variables and then applying stochastic averaging methods with energy envelopes to obtain the Ito differential equations and obtaining the Kolmogorov backward equations (KBE) related to the system energy. Then, combining Monte Carlo sampling to perform data-driven and neural network, a new algorithm is obtained to solve the reliability function that satisfies KBE, which is the innovation of this paper. The algorithm does not need boundary conditions and reduces the need for data volume in high-dimensional problems.