Learning effective dynamics from data-driven stochastic systems

Numerous complex systems in the areas of science, engineering, chemistry or material science have the philosophy of multiscale properties in their dynamic evolution [1-4]. By considering models at different scales simultaneously, we would like to obtain both the efficiency of the macroscopic models as well as the accuracy of the microscopic models. For example, approaches in chemistry usually involve the quantum mechanics models in the reaction region and the classical molecular models elsewhere [5]. Besides, as noisy observations always exist in all kinds of systems under internal or external factors, stochastic dynamical systems come to play an important role in modeling such phenomena. Thus, it is of great importance to study multiscale stochastic dynamical systems [5, 6]. To better understand the intrinsic nature of such complex systems, researchers usually try to investigate the effective dynamics of these systems, such as invariant manifolds, global attractors, tipping points, noise induced bifurcations, transition pathways, and so on [7-11]. These dynamical behaviors could capture the fundamental structures when the system evolves over time or parameter space.