Data-driven Analysis For A Class Of Non-gaussian Stochastic Dynamics

This thesis aims to study the underlying non-Gaussian stochastic dynamical systems based on observation data by combining deep learning with stochastic dynamical systems.There are three types of questions that are considered,that is,extracting stochastic governing laws,discovering transition phenomena,and deriving a reduced model of stochastic dynamical systems.The contents of this thesis are as follows:Chapter 1 presents the research background of data-driven predictive analysis for stochastic dynamics.Chapter 2 reviews some mathematical background knowledge that is relevant to the problems considered in this thesis.This includes some basic concepts of stochastic dynamical systems and deep learning techniques.Chapter 3 is about how to discover non-Gaussian stochastic dynamical system transition phenomena from data.By contacting the Koopman operator with the infinitesimal generator of a stochastic differential equation,the mean exit time and escape probability of a stochastic dynamical system are learned from data.To this end,the finite-dimensional approximation of the Koopman operator is obtained by using the extended dynamic mode decomposition algorithm,and then the finite-dimensional approximation of the infinitesimal generator is obtained.Subsequently,the finite-dimensional matrix approximation of the infinitesimal generator is used to learn the stochastic differential equation as the governing law.Finally,the mean exit time and escape probability are estimated.Numerical results show that this method can be used to discover the transition phenomena of stochastic differential equations with Lévy motion.Chapter 4 investigates how to extract stochastic differential equations with Lévy motions as governing laws from data.To be specific,the non-local Kramers-Moyal formulas establish the relationship between the transition probability density and the coefficients(drift coefficient,diffsusion coefficient,and jump measure)of a stochastic differential equation.Together with deep learning techniques,the transition probability density is estimated from data.Subsequently,the coefficients of the stochastic differential equation are learned.The validity of the method is verified by numerical experiments,and the derivation of the nonlocal Kramers-Moyal formulas is also given.Compared with Chapter 3,a more general stochastic differential equation with Lévy motion is considered here.Chapter 5 studies how to estimate the evolution of transition probability density from sample path data.First,the time information is explicitly embedded into the normalizing flows(a kind of deep learning generation model,which can be used for probability density estimation),and then the estimation of the transition probability density is obtained by minimizing the negative log-likelihood function.Second,the estimated results are compared with the solution of the Fokker-Planck equation.Finally,the robustness of the proposed approach is verified by investigating several multi-dimensional stochastic differential equations,including multimodal distributions and multiplicative noise cases.Chapter 6 presents an effective model reduction method of slow-fast stochastic dynamical systems.To be specific,a deep learning model is used to learn the evolution of the transition probability density of the slow variable from observation data.As a result,the effective reduced model in the probability measure space is obtained.Finally,the validity of this approach is verified by numerical experiments.Chapter 7 provides a summary of this thesis and discusses the pros and cons of these approaches.In the end,the future research topics are discussed.