Nonparametric sparse learning of dynamical systems

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Electrical & Computer Engr

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Hou, Boya

Contributors dc:contributor

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

• Dynamical System
• Statistical Inference, Sparse Learning
• Reproducing Kernel Hilbert Space

Statement dc:rights

• Copyright 2024 Boya Hou

Language dc:language

eng, en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/127331