Analysis of stochastic dynamic responses based on the reduced-dimensional probability evolution equation under the influence of additive Gaussian white noise

The solution of derivative moments in the reduced-dimensional probability evolution equation (RDPEE) remains a challenging problem in contemporary research, particularly for complex systems.This paper introduces the Cell Renormalization Method (CRM), which dissects both the state and probabilistic spaces, and applies it for the numerical calculation of the derivative moments at cell-centered coordinates.Subsequently, an efficient nonlinear regression technique, grounded in Bayesian inference and known as Gaussian Process Regression (GPR), is employed to obtain the continuous curve of derivative moments within the state space.Finally, by solving the RDPEE using the path integral approach, the final probability distribution can be easily recovered.Numerical examples demonstrate the efficacy of the proposed method.