Graduation Year
2025
Document Type
Dissertation
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
Joel Rosenfeld, Ph.D.
Committee Member
Sherwin Kouchekian, Ph.D.
Committee Member
Evguenii Rakhmanov, Ph.D.
Committee Member
Catherine Beneteau, Ph.D.
Committee Member
John Murray-Bruce, Ph.D.
Keywords
machine learning, dynamic systems, occupation kernels, Liouville operator
Abstract
System identification is the process of determining mathematical models that describe the dynamics of a system from data. Dynamic Mode Decomposition (DMD) and Sparse Identification of Nonlinear Dynamical Systems (SINDy) are two distinct approaches used for this purpose.
DMD identifies dominant spatiotemporal modes and eigenvalues that describe the evo lution of a system. It assumes a near-linear representation of dynamics and is closely linked to the Koopman operator, making it ideal for analyzing fluid flows, oscillatory systems, and modal structures. The DMD method uses time series data where each data point is referred to as a snapshot and represents the value of the state at a particular moment in time.
SINDy discovers explicit differential equations that govern the system’s behavior by using sparse regression. It selects a minimal set of relevant nonlinear terms from a candidate function library, making it powerful for learning governing equations in complex systems.
This project aims to explore different methods of system identification. In the first project, we use the weighted composition operator (WCO) on RKHS, Here we consider the compactness and boundedness properties of WCO on Reproducing Kernel Hilbert space (RKHS) that gives it an edge over the popular Koopman operator which is often used in Dynamic mode Decomposition but is not necessarily compact and bounded. We first represent the WCO as a finite rank operator, then perform a reconstruction of the system and compare accuracy in prediction against the Koopman-based KDMD method and consider the effect of time-delay on it’s reconstruction.
The second project, partially inspired by the SINDy algorithm sees a definition of a new inner product using the properties of the occupational kernel on the Liouville operator. This inner product helps us construct a closed dynamical system of equations to describe the evolution of the slow dynamics of a system using the Mori-Zwanzig formalism.
Finally, using the weighted composition operator, we study DMD of control affine dy namical systems using control kernels. In this project, The separation of drift dynamics from input dynamics was achieved through the application of the weighted composition operator and control kernels. A multiplication operator was introduced as the feedback controller, fa cilitating the control mechanism. Subsequently, the composition of the control-weighted com position operator and the multiplication operator was formulated as a finite-rank operator. Through spectral decomposition, the Dynamic Mode Decomposition (DMD) methodology was employed to predict the system’s snapshots. This process was implemented using both the eigenfunction-based approach and the Singular Value Decomposition (SVD) method.
Scholar Commons Citation
Amagwula, Chukwuebuka, "Exploring System Identification of Non-Linear Dynamics using the Weighted Composition operator and the Liouville operator" (2025). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/10841
