DR-PDEE: A unified computational framework for high-dimensional nonlinear and non-Markov stochastic dynamical systems under general nonstationary excitations

The analysis of high-dimensional nonlinear stochastic dynamical systems remains a central challenge in computational structural dynamics, particularly when dealing with non-Markov processes and non-Gaussian, nonstationary, or discontinuous excitations. The recently developed dimension-reduced probability density evolution equation (DR-PDEE) provides a physics-driven and mathematically rigorous framework that enables efficient and accurate probabilistic characterization of such complex systems. Originating from the principle of preservation of probability, the DR-PDEE constructs an exact lower-dimensional governing equation for the transient probability density function (PDF) of stochastic responses, without relying on Markovian or Gaussian white-noise approximations. The formulation closes the reduced partial differential equation through the introduction of an intrinsic drift function – defined as the conditional mean of the original high-dimensional drift – thereby overcoming the curse of dimensionality while retaining full probabilistic fidelity. This paper systematically examines the theoretical foundation and computational realization of DR-PDEE for nonlinear, non-Markov, and path-discontinuous systems subjected to nonstationary, non-Gaussian, and non-white excitations, including Poisson and multiplicative stochastic processes. Representative numerical examples demonstrate the method’s robustness and accuracy compared with traditional Monte Carlo simulations. The results highlight the DR-PDEE’s potential as a unified, data-compatible computational paradigm for stochastic engineering dynamics, bridging the gap between physics-based modeling and data-driven inference.