Author: Weize Mao

Publisher:

Published: 2021

Total Pages: 118

ISBN-13:

In this dissertation, we discuss several topics in data-driven learning of unknown dynamical systems with missing information. Depending on different scenarios of the underlying governing equations and data collection, we need different learning techniques, in order to effectively learn the underlying dynamics. The dissertation consists of five chapters. In the first two chapters, we review basic techniques regarding regression, neural networks, and data-driven learning of dynamical systems. In each of the last three chapters, we introduce data-driven learning method in different scenarios of underlying dynamical systems and data. First, we assume the underlying dynamical system is autonomous and contains unknown parameters, and we only have access to statistical moments (e.g. mean) of the measurements on the state variables. When a variable is Markovian (i.e. memoryless), deep learning methods such as \cite{qin2018data} can be readily applied to learn the time evolution of the variable. In our case, even though the underlying dynamical system is autonomous, the resulting moments are not Markovian, hence new methods need to be developed to corporate the memory effect. The flow map governing the moments time evolution with memory is derived based on Mori-Zwanzig formalism and is approximated by our proposed residual network. Second, in some cases, we have measurement data for individual trajectories of the state variables. When we have access to parameter information, data-driven methods such as \cite{QinCJX_IJUQ20} are effective at recovering the underlying parameterized system. But when information about the parameters is missing, new methods need to be developed. By treating the parameters as missing state variables with zero derivatives, the observed variables are essentially a reduced system, which has memory. We then develop a new method that incorporates the memory effect for learning the reduced system. Lastly, we are interested in recovering the dynamics of species populations in chemical reactions, using observational data. The evolution of species populations is a stochastic process whose probability distribution is governed by \textit{chemical master equation} (CME), which is a set of ordinary differential equations (ODEs). The CME consists of a large number of variables, and is intractable to be solved directly. The moments equations are derived based on CME, and we develop a data-driven method to learn the time evolution of the moments.