Data-driven method for dimension reduction of nonlinear randomly vibrating systems

Data-driven identification of nonlinear differential equations turns out to be an inefficient, and sometimes even impossible, for high-dimensional randomly vibrating systems. The mathematical formulism of stochastic averaging, moreover, is restricted by the difficulty of determining slowly-varying processes and the complexity of deriving the drift and diffusion coefficients. This paper is devoted to the dimension reduction of nonlinear randomly vibrating systems by merging the data-driven identification method and the idea of stochastic averaging. The slowly-varying processes (i.e., the invariants of the associated conservative systems) are first identified in the framework of Koopman operator theory, and then the drift and diffusion coefficients are derived from discrete data by the sparse optimization. The proposed method retains the advantages at the same time avoids the disadvantages of the data-driven identification methods and stochastic averaging, and can be regarded as the data-driven version of stochastic averaging. The application and efficacy of this method are demonstrated by four specific numerical examples, including a non-smooth oscillator and a two-degree-of-freedom nonlinear system.