A new solution method for high-dimensional stochastic dynamical systems via delay embedding

Stochastic phenomena are ubiquitous in natural and engineered systems, and solving high-dimensional stochastic dynamical systems remains a fundamental challenge. In many practical applications, however, only a subset of the system variables is of primary interest. This motivates the development of effective dimensionality-reduction strategies for analyzing such complex systems. In this study, we propose a data-driven computational framework for high-dimensional stochastic dynamical systems based on the concept of delay embedding. The proposed approach constructs a family of delay-embedding mappings from multiple time series of the original system and incorporates these mappings into the governing equations. As a result, the original high-dimensional system is approximately transformed into a low-dimensional time-delay system. The performance of the framework is evaluated through probability density function analysis of two four-dimensional systems and one ten-dimensional system under Gaussian white noise excitation. Numerical results demonstrate excellent agreement with Monte Carlo simulations while achieving substantial reductions in computational cost. These findings indicate that the proposed framework provides an efficient and accurate tool for the analysis of high-dimensional stochastic systems.