Nonparametric sparse learning of dynamical systems

Hou, Boya

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https://hdl.handle.net/2142/127331 Copy

Description

Title

Nonparametric sparse learning of dynamical systems

Author(s)

Hou, Boya

Issue Date

2024-10-21

Director of Research (if dissertation) or Advisor (if thesis)

Bose, Subhonmesh

Doctoral Committee Chair(s)

Bose, Subhonmesh

Committee Member(s)

• Basar, Tamer
• Raginsky, Maxim
• Srikant, Rayadurgam
• Vaidya, Umesh

Department of Study

Electrical & Computer Eng

Discipline

Electrical & Computer Engr

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Dynamical system
• statistical inference, sparse learning
• reproducing kernel Hilbert space

Abstract

Transfer operators such as the Koopman operator and the Perron-Frobenius operator provide a rich framework for representing the dynamics of very general, nonlinear dynamical systems. In this dissertation, we develop a nonparametric approach to learning the system dynamics via transfer operators in reproducing kernel Hilbert spaces (RKHS). Compared with methods using fixed parametric structures, the proposed nonparametric representation does not require manually engineered features, and the model grows and adjusts with the amount of training data, making it appealing from a data-oriented standpoint. As the Koopman operator governs the evolution of function, its ability to capture system dynamics depends on the selection of the function space. A major limitation of existing literature is that they often require the chosen function space to be closed under the system dynamics. In general, this closedness assumption is quite restrictive as the dynamics might escape the chosen function space. To address this challenge, we investigate the ``mis-specified'' scenario where the Koopman operator maps a function in RKHS to a larger space of equivalent classes of functions, and this space is generated by an eigensystem associated with an integral operator. In this respect, we view our work as a first step toward understanding the mathematical consequences of mis-specification in learning the operator. When interacting with reproducing kernel Hilbert spaces (RKHS), descriptions of dynamics often incur prohibitive data storage requirements. To counter this difficulty, we explore techniques to reduce redundancy in datasets as a precursory step to computation. This sparsification step yields simpler system representations with lower data storage requirements. In addition, we provide theoretical studies on how the relationship between the number of samples and the resulting performance can be controlled. For the task of data-driven learning and decision-making, it is important to understand how much data is required to construct an accurate model of the unknown nonlinear system dynamics. What makes learning dynamical systems more challenging than classic statistical estimation is that data is typically available in the form of trajectories, which breaks the key independence assumption underlying state-of-the-art machine learning methods. For the proposed operator-theoretic system identification framework, we establish finite-sample complexity guarantees in the batch learning scenario and last-iterate convergence guarantees in the online case with streaming samples. To illustrate the efficacy of our sparse learning approach, we empirically examine the algorithm through deterministic and stochastic nonlinear system examples. We further integrated this approach into model-based optimal control designs, power system transient stability analysis, and uncertainty propagation through unknown nonlinear system dynamics.

Graduation Semester

2024-12

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/127331

Copyright and License Information

Copyright 2024 Boya Hou

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