Universal Runge–Kutta Neural Oscillator for Stochastic Response Analysis of Nonlinear Dynamic Systems under Random Loads

Abstract

The Runge–Kutta neural network effectively captures the physical characteristics of dynamic structures, offering clear interpretability and making it well-suited for structural response analysis. However, this neural network architecture assumes that target function-to-function mapping relationships can be explicitly described by neural ordinary differential equations. This strict assumption limits the neural network architecture’s universality. This study proposes a universal neural oscillator constructed by augmenting a second-order neural ordinary differential equation with a feedforward neural network and its time-discretized counterpart, a universal Runge–Kutta neural oscillator. The proposed neural oscillator proves to be universal for the causal continuous operators between continuous function spaces. Besides, it also retains the capability to learn the underlying physical equations that govern the target function-to-function relationships if they exist. The effectiveness and versatility of the proposed neural oscillator for stochastic response analysis of nonlinear dynamic systems are validated through three numerical cases involving various nonlinear systems. The results demonstrate that the proposed Runge–Kutta neural oscillator significantly enhances the versatility of the original Runge–Kutta neural network and it is effective in estimating both the probability distribution and the extreme value distribution of the response processes for complex nonlinear dynamic systems under random loads, as well as in learning the underlying physical equations.