This project trains a neural ordinary differential equation (neural ODE) on real experimental data to learn the nonlinear damping law of a mass–spring system, and compares the learned dynamics against the classical linear-damping model taught in an introductory differential equations course.
It is the companion code for the paper:
N. Albin, A. G. Bennett, and A. Chand. "Machine Learning for Modeling in an Elementary Differential Equations Class." CODEE Journal, vol. 20, no. 2, 2026. scholarship.claremont.edu/codee/vol20/iss2/1 · DOI: 10.5642/codee.QKJJ1808
Warning
To reset this project for students, you'll need to find the seed choices in the section Fitting the model and swap the commented-out line.
A standard mass–spring model assumes linear damping, m x'' + c x' + k x = 0.
Real systems often damp nonlinearly. Instead of guessing a functional form, this
project represents the damping term with a small neural network and learns it
directly from measured trajectory data by:
- casting the dynamics as a first-order ODE system,
- parameterizing the unknown (nonlinear) damping with a neural network,
- integrating the system with a differentiable ODE solver, and
- fitting the network by gradient descent so the simulated trajectory matches the experimental data.
The notebook also discusses when machine learning helps and when it does not — how much data is needed, and how the learned model compares to the linear baseline.
| File | Description |
|---|---|
Learning_a_Nonlinear_Damping_Model.ipynb |
Main notebook: data loading, model definition, training, and comparison plots. Runs as-is on Google Colab — no local setup required. |
data.csv |
Experimental trajectory data for the mass–spring system. |
model.pt |
Trained PyTorch model weights. |
Easiest — Google Colab: open the notebook in Colab and run all cells; no installation needed.
Locally:
pip install -r requirements.txt
jupyter lab Learning_a_Nonlinear_Damping_Model.ipynb
Python · PyTorch · neural ODEs · Jupyter
MIT — see LICENSE.