DR-PDEE for engineered high-dimensional nonlinear stochastic systems: a physically-driven equation providing theoretical basis for data-driven approaches

For over half a century, the analysis, control, and optimization design of high-dimensional nonlinear stochastic dynamical systems have posed long-standing challenges in the fields of science and engineering. Emerging scientific ideas and powerful technologies, such as big data and artificial intelligence (AI), offer new opportunity for addressing this problem. Data-driven techniques and AI methods are beginning to empower the research on stochastic dynamics. However, what is the physical essence, theoretical foundation, and effective applicable spectrum of data-driven and AI-aided (DDAA) stochastic dynamics? Answering this question has become important and urgent for advancing research in stochastic dynamics more solidly and effectively. This paper will provide a perspective on answering this question from the viewpoint of system dimensionality reduction. In the DDAA framework, the dimension of observed data of the studied system, such as the dimension of the complete state variables of the system, is fundamentally unknown. Thus, it can be considered that the stochastic dynamical systems under the DDAA framework are dimension-reduced subsystems of real-world systems. Therefore, a question of interest is: To what extent can the probability information predicted by the dimension-reduced subsystem characterize the probability information of the real-world system and serve as a basis for decision-making? The paper will discuss issues such as the dimension-reduced probability density evolution equation (DR-PDEE) satisfied by the probability density function (PDF) of path-continuous non-Markov responses in general high-dimensional systems, the dimension-reduced partial integro-differential equation satisfied by the PDF of path-discontinuous responses, and the non-exchangeability of dimension reduction and imposition of absorbing boundary conditions. These studies suggest that the DR-PDEE and the dimension-reduced partial integro-differential equation can serve as important theoretical bases for the effectiveness and applicability boundaries of the DDAA framework.