Date of Award

May 2022

Degree Type

Thesis

Degree Name

Master of Science

Department

Mathematics

First Advisor

Dexuan Xie

Committee Members

Istvan Lauko, Gabriella Pinter

Keywords

Deep Neural Network, Dynamic Mode Decomposition, Dynamical Systems, Machine Learning, Singular Value Decomposition, Sparse Identification of Nonlinear Dynamics

Abstract

Dynamical Systems are ubiquitous in mathematics and science and have been used to model many important application problems such as population dynamics, fluid flow, and control systems. However, some of them are challenging to construct from the traditional mathematical techniques. To combat such problems, various machine learning techniques exist that attempt to use collected data to form predictions that can approximate the dynamical system of interest. This thesis will study some basic machine learning techniques for predicting system dynamics from the data generated by test systems. In particular, the methods of Dynamic Mode Decomposition (DMD), Sparse Identification of Nonlinear Dynamics (SINDy), Singular Value Decomposition (SVD), and Deep Neural Network (DNN) regression will be studied. Such techniques provide alternatives to determine the dynamics of a system of interest without needing to resort to the computationally expensive elementary methods. From numerically testing a few linear and nonlinear systems of ordinary differential equations, it was observed that the methods of DMD and SVD could approximate linear systems effectively but performed poorly against nonlinear systems. The approach of DNN regression proved effective for both linear and nonlinear dynamical systems.

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