Graduation Year

2023

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Joel A. Rosenfeld, Ph.D.

Committee Member

Gangaram Ladde, Ph.D.

Committee Member

Seung-Yeop Lee, Ph.D.

Committee Member

Lu Lu, Ph.D.

Committee Member

Tansel Yucelen, Ph.D.

Keywords

Dynamic systems, Liouville operator, Machine learning, occupation kernels

Abstract

Dynamic mode decomposition (DMD) is a physically interpretable data driven technique for modeling dynamical systems. 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. The important patterns found in the data are extracted and then used to create a model for the system. Once the important pieces have been obtained reduced order modeling can be used to reduce the complexity of the model.

But, DMD has evolved since its inception. After its association with the Koopman operator, DMD has grown in popularity. New DMD methods and new applications for DMD are constantly being published. Some methods that are of great importance are extended DMD (EDMD), kernel-based DMD (KDMD), and DMD for large and streaming data sets (sDMD). All of these methods are associated with the Koopman operator and are therefore limited in their applicability.

The work presented in this dissertation expands on a DMD method for continuous-time dynamical systems which is called occupation kernel-based DMD (OKDMD). The OKDMD method is associated with the Liouville operator and therefore is not subject to the limitations of prior DMD methods. By exploiting the properties of the Liouville operator and occupation kernels two new methods are presented in this dissertation.

The first method that is introduced is called partial knowledge OKDMD (POKeDMD). In creating the POKeDMD method we took the view point that if information on the dynamics of a system is available, it should be used to aid the data driven method in the learning process. The POKeDMD method uses known information about the dynamics of the system in order to learn a model for the unknown portion of the dynamics. The known dynamics plus the model created for the unknown dynamics should produce a model that better fits the data.

The second method is called streaming OKDMD (StOKeDMD). The method expands the usability of OKDMD to systems with a large number of snapshots and allows for the use of streaming data. Another advantage of the StOKeDMD method is that subsampling of large data or streaming data sets is not necessary. Being able to avoid the subsampling of our data should lead to better models.

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