Date of Award
12-2024
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics and Statistics
First Advisor
Xiaojing Ye
Abstract
We investigate the usefulness of deep learning when applied to both control theory and partial differential equations (PDEs). We will develop new network architectures and methodologies to approach the solving of high-dimensional problems. Specifically, we develop a network architecture called Lyapunov-Net for approximating Lyapunov functions in high-dimensions and a new methodology called Neural Control for finding solution operators for high-dimensional parabolic PDEs. The theoretical accuracy and numerical efficiency of these approaches will be investigated along with implementation details to use them in practice.
Recommended Citation
Gaby, Nathan, "Neural Networks and Approximation of High-dimensional Functions: Applications in Control and Partial Differential Equations." Dissertation, Georgia State University, 2024.
https://scholarworks.gsu.edu/math_diss/98
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