Usually learning dynamical systems by data-driven methods requires large amount of training data, which may be time consuming and expensive. Active learning, aims at choosing the most informative samples to make more efficient is a promising way solve this issue. However, actively difficult since it not possible arbitrarily sample state-action space under constraint system dynamics. The state-o...

Motivated by the recent work of Benjamini, Häggström, Peres, and Steif (2003) on dynamical random walks, we: (i) Prove that, after a suitable normalization, the dynamical Gaussian walk converges weakly to the Ornstein–Uhlenbeck process in classical Wiener space; (ii) derive sharp tailasymptotics for the probabilities of large deviations of the said dynamical walk; and (iii) characterize (by way...

‎In this paper we investigate the dilations of completely positive definite representations‎ ‎of (C^ast)-dynamical systems with abelian groups on Hilbert (C^ast)-modules‎. ‎We show that if ((mathcal{A}‎, ‎G,alpha)) is a (C^ast)-dynamical system with (G) an abelian group‎, ‎then every completely positive definite covariant representation ((pi,varphi,E)) of ((mathcal{A}‎, ‎G,alpha)) on a Hilbert ...

As a model for multiscale systems under random influences on physical boundary, a stochastic partial differential equation under a fast random dynamical boundary condition is investigated. An effective equation is derived and justified by reducing the random dynamical boundary condition to a random static boundary condition. The effective system is still a stochastic partial differential equati...

Invariant manifolds facilitate the understanding of nonlinear stochastic dynamics. When an invariant manifold is represented approximately by a graph for example, the whole stochastic dynamical system may be reduced or restricted to this manifold. This reduced system may provide valuable dynamical information for the original system. The authors have derived an invariant manifold reduction or r...