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Understanding and Prediction of Dynamical Systems with Machine Learning.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Understanding and Prediction of Dynamical Systems with Machine Learning./
Author:	Hou, Xiao.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2021,
Description:	142 p.
Notes:	Source: Dissertations Abstracts International, Volume: 83-02, Section: B.
Contained By:	Dissertations Abstracts International83-02B.
Subject:	Applied mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28718956
ISBN:	9798538106479

Understanding and Prediction of Dynamical Systems with Machine Learning.
Hou, Xiao.

Understanding and Prediction of Dynamical Systems with Machine Learning. - Ann Arbor : ProQuest Dissertations & Theses, 2021 - 142 p.

Source: Dissertations Abstracts International, Volume: 83-02, Section: B.

Thesis (Ph.D.)--The University of Wisconsin - Madison, 2021.

This item must not be sold to any third party vendors.

Complex nonlinear dynamical systems are frequently utilized and play a significant role in the study of natural science and many other fields. This thesis studies dynamical systems both theoretically and numerically with the help of machine learning, quantifying uncertainties in the system and developing models and algorithms to explore two specific application problems. An information criterion is first introduced to quantify model error of dynamical systems, and then two specific applications of machine learning algorithms on the study of dynamical systems are presented in the context of COVID-19 pandemic modeling and partial differential equation (PDE) solvers, respectively.The first part of this thesis introduces a simple information criterion that is developed to quantify the model error in imperfect models, incorporating not only the equilibrium statistics but also temporal information of the system. Three applications of this information criterion are presented, including the study of model reduction, stochastic parameterizations and intermittent events, all of which are illustrated with detailed examples. The second part of this thesis investigates a specific example of the combination of machine learning algorithms and dynamical systems in the context of COVID-19 pandemic in 2020. A new human mobility flow-augmented stochastic SEIR model is developed, which employs machine learning community detection method, data assimilation technique, and online learning of parameters with a stochastic parameterization approach. Numerical simulations are performed to reconstruct the historical transmission trajectories of COVID-19 in the two largest counties in Wisconsin, and the results are further analyzed to inspect the associations between COVID-19 transmission and human mobility as well as demographic features. Several scenario studies are performed during the study, providing insights for policymakers on the design of effective regionalization-based policies to control the ongoing spread of COVID-19. The reliability of community detection results and the validity of the dynamical model are further inspected by conducting numerous sensitivity tests and building several new models for comparison.As the last work of this thesis, a new numerical algorithm is proposed and analyzed in order to solve hyperbolic conservation laws. This newly designed numerical scheme combines the Lax Wendroff method with $l_1$ regularization and, comparing to past work with similar techniques, adds a new critical conservation constraint. It is proved that this scheme is consistent, convergent, total variation diminishing, and satisfies the weak entropy condition for conservation laws. Numerical solutions to Burgers and Euler's equation are presented to validate the analytical results and to explore possible improvement to the algorithm.

ISBN: 9798538106479Subjects--Topical Terms:

2122814
Applied mathematics.
Subjects--Index Terms:

Dynamical systems

Understanding and Prediction of Dynamical Systems with Machine Learning.
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506 $a This item must not be sold to any third party vendors.
520 $a Complex nonlinear dynamical systems are frequently utilized and play a significant role in the study of natural science and many other fields. This thesis studies dynamical systems both theoretically and numerically with the help of machine learning, quantifying uncertainties in the system and developing models and algorithms to explore two specific application problems. An information criterion is first introduced to quantify model error of dynamical systems, and then two specific applications of machine learning algorithms on the study of dynamical systems are presented in the context of COVID-19 pandemic modeling and partial differential equation (PDE) solvers, respectively.The first part of this thesis introduces a simple information criterion that is developed to quantify the model error in imperfect models, incorporating not only the equilibrium statistics but also temporal information of the system. Three applications of this information criterion are presented, including the study of model reduction, stochastic parameterizations and intermittent events, all of which are illustrated with detailed examples. The second part of this thesis investigates a specific example of the combination of machine learning algorithms and dynamical systems in the context of COVID-19 pandemic in 2020. A new human mobility flow-augmented stochastic SEIR model is developed, which employs machine learning community detection method, data assimilation technique, and online learning of parameters with a stochastic parameterization approach. Numerical simulations are performed to reconstruct the historical transmission trajectories of COVID-19 in the two largest counties in Wisconsin, and the results are further analyzed to inspect the associations between COVID-19 transmission and human mobility as well as demographic features. Several scenario studies are performed during the study, providing insights for policymakers on the design of effective regionalization-based policies to control the ongoing spread of COVID-19. The reliability of community detection results and the validity of the dynamical model are further inspected by conducting numerous sensitivity tests and building several new models for comparison.As the last work of this thesis, a new numerical algorithm is proposed and analyzed in order to solve hyperbolic conservation laws. This newly designed numerical scheme combines the Lax Wendroff method with $l_1$ regularization and, comparing to past work with similar techniques, adds a new critical conservation constraint. It is proved that this scheme is consistent, convergent, total variation diminishing, and satisfies the weak entropy condition for conservation laws. Numerical solutions to Burgers and Euler's equation are presented to validate the analytical results and to explore possible improvement to the algorithm.
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