Date of Award
Summer 7-22-2026
Document Type
Thesis
Publication Status
Version of Record
Submission Date
August 2026
Department
Mathematical Sciences
College Granting Degree
Charles E. Schmidt College of Science
Department Granting Degree
Mathematics
Degree Name
Master of Science (MS)
Thesis/Dissertation Advisor [Chair]
Jason Mireles-James
Abstract
Neural Ordinary differential equations(Neural ODEs) provide a continuous-time framework for learning dynamical systems by representing the evolution of a state variable through a parameterized differential equation. While most existing methods rely on numerical time stepping methods during training such as RK4, this work investigates an alternate spectral framework where both the solution and the time-dependent network parameters are represented by global polynomial expressions.
Two methods are employed: the first uses Taylor series representations, and the second approach uses Chebyshev polynomial expansions together with fast spectral transforms and efficient polynomial multiplication techniques. In each formulation, the Neural ODE is transformed into a finite-dimensional nonlinear algebraic system whose unknowns are the spectral coefficients of the state, weights, and biases. The resulting optimization problem is then solved by gradient based methods to minimize the discrepancy between the computed solution and a prescribed target function.
The proposed methods are evaluated on several benchmark targets exhibiting a range of analytic behavior, which includes exponential functions, Runge-type rational functions, and functions possessing essential singularities. Numerical experiments compare the approximation quality, convergence behavior, and scalability of the Taylor and Chebyshev Formulations. Although the Taylor approach provides a straightforward implementation, its computational cost grows rapidly due to repeated Cauchy-product operations. In contrast, the Chebyshev formulation leverages fast-transform-based algorithms, enabling substantially higher polynomial degrees while maintaining accuracy and improving computational efficiency.
These results demonstrate that spectral representations may provide a viable alternative framework for training Neural ODEs and suggest that Chebyshev-based methods, in particular, offer significant advantages for high-order approximations and large-scale computations.
Recommended Citation
Brammer, Andreus, "A DYNAMICS VIEWPOINT ON THE APPROXIMATION CAPABILITIES OF INFINITE-DEPTH NEURAL NETWORKS" (2026). Electronic Theses and Dissertations. 377.
https://digitalcommons.fau.edu/etd_general/377