First Advisor

Bruno Jedynak

Term of Graduation

Spring 2025

Date of Publication

5-6-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences

Department

Mathematics and Statistics

Language

English

Subjects

Dynamical Systems, Finite Elements, Machine Learning, Partial Differential Equations, Variational Forms

Physical Description

1 online resource (xi, 182 pages)

Abstract

We combine numerical and machine learning techniques to present a general framework for solving inverse problems using vector valued reproducing kernel Hilbert spaces in a variational formulation. We present this framework in two papers. In the first paper, we present an original state-of-the-art method derived in the context of our general framework for learning dynamical systems. In the second paper, we generalize the method from our first paper to arrive at the framework for solving inverse problems. Then we apply our general framework to the task of learning dynamical systems. In both papers we consider numerous applications of our methods and demonstrate the effectiveness of our methods with experiments on datasets from dynamical systems. Our experiments demonstrate our methods are both computationally efficient and provide compelling performance on the task of learning dynamical systems.

Rights

©2025 Victor William Rielly

In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).

Comments

The work at Portland State University was partly funded by the National Institute of Health RO1AG021155, R01EY032284, R01AG027161, and National Science Foundation #2136228.

Persistent Identifier

https://archives.pdx.edu/ds/psu/43785

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