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
Tansel Yucelen, Ph.D.
Committee Member
Dmitry Khavinson, Ph.D.
Committee Member
Seung-Yeop Lee, Ph.D.
Keywords
Fractional Dynamic Mode Decomposition, Fractional Liouville Operator, Occupation Kernel Hilbert Space, Fractional Dynamical Systems Control, Nonlinearity learning through Kolmogorov-Arnold Network, Stability Analysis for Fractional Kolmogorov-Arnold Network
Abstract
The rapidly growing interest in data-driven modeling and fractional-order systems highlights a keychallenge in modern science and engineering: capturing long-memory effects and nonlinear behaviors in real-world dynamical processes. Conventional integer-order methods, while powerful, often overlook the history-dependent nature of many phenomena—ranging from viscoelastic materials to anomalous diffusion and hereditary feedback systems. This dissertation addresses that gap by blending fractional calculus with cutting-edge machine learning to create robust, memory-aware modeling and control frameworks.
Beginning with an extension of Dynamic Mode Decomposition (DMD) to fractional-order systems, we introduce Fractional Dynamic Mode Decomposition (F-DMD) as a data-driven tool that leverages Mittag- Leffler functions and fractional discretizations. By accommodating power-law relaxation and nonlocal behaviors, F-DMD significantly broadens the applicability of DMD-based Koopman operator methods. We then augment fractional-order modeling with Occupation Kernel Regression (OKR) to handle nonlinearities in both memory and state-space dynamics, offering a versatile system identification approach for incomplete or noisy datasets.
Next, we shift attention to control and develop a new paradigm where Kolmogorov-Arnold Networks (KANs)—which feature trainable activation functions—are explicitly coupled with fractional dynamics. Dubbed fractional KANs, these architectures enable precise control of memory-dependent processes while retaining interpretability. We provide theoretical guarantees on the existence, uniqueness, and stability of solutions, confirmed through comprehensive numerical experiments and a practical mobile robot path-tracking application.
Finally, we analyzed Fractional Kolmogorov-Arnold Network (fKAN) equipped with trainable, fractionalorthogonal Jacobi basis functions to tackle time-delay systems. We present a rigorous Lyapunov-based stability analysis that underscores the resilience of this memory-rich framework under parametric uncertainty and delayed feedback. The strategies and theoretical insights presented here establish a new foundation. This foundation supports reliable, efficient data-driven techniques. These techniques can model, control, and analyze complex systems. Such systems often show deep memory effects. They also involve nonlinear interactions. This work opens promising paths for future studies. Potential areas include machine learning and control theory. Applications may extend to other fields as well.
Scholar Commons Citation
Chen, Haowei(Alice), "Fractional Calculus Approach For Learning Unknown Dynamic System" (2025). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/10929
