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

Stochastic phenomena are pervasive in natural and engineering systems, andthe solution of high-dimensional stochastic dynamics remains a fundamentalchallenge. In many cases, only a subset of system variables is of primary interest. Hence, dimensionality reduction methods provide a viable means to studysuch high-dimensional systems. In this study, we propose a data-driven computational framework for high-dimensional stochastic dynamical systems basedon the concept of delay embedding. The method constructs a family of delayembedding mappings from multiple time series of the original system and incorporates them into the governing equations, thereby approximately transformingthe high-dimensional dynamics into a low-dimensional time-delay system. Theeffectiveness of the proposed framework is demonstrated through probability density function analysis of two four-dimensional systems and one ten-dimensionalsystem subjected to Gaussian white noise excitation. Numerical results show thatthe method achieves accurate agreement with Monte Carlo simulations while substantially reducing computational cost, offering a powerful tool for the efficientanalysis of high-dimensional stochastic systems.