Coupled-Pendulum Modeling in an ODE Class: An Assignment on Fourier-Initialized Gradient Descent in Machine Learning

Authors

• Huy Truong, Kansas State UniversityFollow
• Andrew Bennett, Kansas State UniversityFollow

Publication Date

7-3-2026

Keywords

Machine learning, Coupled pendulums, Fourier analysis, Gradient descent, Differential equations education, Jupyter Notebook

Disciplines

Mathematics | Physical Sciences and Mathematics | Science and Mathematics Education

Abstract

As data-driven methods are increasingly used in science and engineering, students benefit from learning to integrate machine learning techniques with traditional mathematical modeling. We present a hands-on extra-credit assignment for an undergraduate ordinary differential equations (ODE) course that enables students to compare classical analytical methods with data-driven approaches on the same physical system. Using a coupled-pendulum system---two pendulums connected by a spring---with real experimental data acquired via video tracking of a real physical setup, students work through three models in a guided Jupyter notebook with all code provided. First, they fit a neural network with Fourier features as a purely data-driven approach and observe its failure to extrapolate beyond the training data. Second, they fit a physics-based two-frequency solution derived from the coupled ODE using gradient descent, beginning from random initial guesses in order to expose a concrete failure mode of gradient-based optimization. Third, they apply a hybrid approach in which two Fourier-peak frequencies extracted from the data serve as initial guesses for gradient descent, demonstrating how importing mathematical structure into the optimization step can overcome that failure. We find that Fourier-based initialization improves the reliability of gradient-based fitting and yields a strong data fit. The assignment concludes with a transfer task in which students apply these ideas to a new engineering scenario and argue for the approach they would choose.

Recommended Citation

Truong, Huy and Bennett, Andrew (2026) "Coupled-Pendulum Modeling in an ODE Class: An Assignment on Fourier-Initialized Gradient Descent in Machine Learning," CODEE Journal: Vol. 20: Iss. 2, Article 8.
Available at: https://scholarship.claremont.edu/codee/vol20/iss2/8