Random vibrations occur in many engineering systems including buildings subject to earthquake excitation, vehicles traveling on a rough road and off-shore platform in random waves. Analysis of random vibrations for linear systems has been well studied. For nonlinear systems, particularly for multi-degree-of-freedom systems, however, there are still many challenges including analyzing the probability distribution of transient responses of the system. Monte Carlo simulation was considered the only viable method for this task. In this paper, We propose a method to construct semi-analytical transient solutions of the probability distribution of transient responses of nonlinear systems by using the radial basis function neural networks. The activation functions consist of normalized Gaussian probability density functions. Two examples are presented to show the effectiveness of the proposed solution method. The transient probability distributions and response moments of these examples are presented, which have not been reported in the literature before.

The first-passage time probability plays an important role in the reliability assessment of dynamic systems in random vibrations. To find the solution of the first-passage time probability is a challenging task. The analytical solution to this problem is not available even for linear dynamic systems. For nonlinear and multi-degree-of-freedom systems, it is even more challenging. This paper proposes a radial basis function neural networks method for solving the first-passage time probability problem of linear, nonlinear, and multi-degree-of-freedom dynamic systems. In this paper, the proposed method is applied to solve for the backward Kolmogorov equation subject to boundary conditions defined by the safe domain. A null-space solution strategy is proposed to deal with the boundary condition. Several examples including a two degrees-of-freedom nonlinear Duffing system are studied with the proposed method. The results are compared with Monte Carlo simulations. It is believed that the radial basis function neural networks method provides a new and effective tool for the reliability assessment and design of multi-degree-of-freedom nonlinear stochastic dynamic systems.

This paper presents a novel method for analyzing high-dimensional nonlinear stochastic dynamic systems. In particular, we attempt to obtain the solution of the Fokker–Planck–Kolmogorov (FPK) equation governing the response probability density of the system without using the FPK equation directly. The method consists of several important components including the radial basis function neural networks (RBFNN), Feynman–Kac formula and the short-time Gaussian property of the response process. In the area of solving partial differential equations (PDEs) using neural networks, known as physics-informed neural network (PINN), the proposed method presents an effective alternative for obtaining solutions of PDEs without directly dealing with the equation, thus avoids evaluating the derivatives of the equation. This approach has a potential to make the neural network-based solution more efficient and accurate. Several highly challenging examples of nonlinear stochastic systems are presented in the paper to illustrate the effectiveness of the proposed method in comparison to the equation-based RBFNN approach.