This repository contains reproducible code examples for the NSF CBMS 2025: Computational Mathematics and AI course—a ten-lecture introductory series on research topics at the intersection of computational mathematics and artificial intelligence. The course explores how computational mathematics provides foundations, precise language, and design principles for AI, and how AI enables new capabilities for tackling previously intractable computational problems.

Course Information:

• Dates: December 8–12, 2025
• Location: Houston, Texas
• Instructor: Lars Ruthotto (Emory University)
• Conference: Research at the Interface of Applied Mathematics and Machine Learning (CBMS-AMML)

Resources:

• Conference Flyer (PDF)
• Lecture Slides
• Lecture Videos (YouTube)

pip install -r requirements.txt

Notebook	Colab	Description
01-polynomial-double-descent.ipynb		Demonstrates the double descent phenomenon in polynomial curve fitting. Includes Picard plot analysis.
03-optimization/peaks_optimization.ipynb		Neural network optimization on 2D peaks classification. Compares SGD, Adam, and TR-GN across small, lazy/NTK, and mean-field regimes.
07-sciml/classical-solver.ipynb		Classical CG solver for 2D Darcy flow. Baseline comparison for neural methods.
07-sciml/pinn.ipynb		Physics-Informed Neural Networks for solving PDEs without training data.
07-sciml/operator-learning.ipynb		Fourier Neural Operator (FNO) and DeepONet for learning solution operators.
08-stochastic-control/stochastic-control.ipynb		100D stochastic optimal control. Compares PINN, FBSNN, and NeuralSOC methods.
09-inverse-problems/dps-gmm.ipynb		Diffusion Posterior Sampling (DPS) for Bayesian inference on 1D GMM.
09-inverse-problems/deblurring-cg.ipynb		Image deblurring with conjugate gradient methods.

Open the notebooks in Jupyter:

jupyter notebook

Or in JupyterLab:

jupyter lab

Each notebook is self-contained and includes all helper functions inline.

• Python 3.9+
• NumPy, SciPy, Matplotlib
• JAX (for the JAX version)

See requirements.txt for full dependencies.

This conference is supported under NSF CBMS Award Number 2430460 and by the Department of Mathematics at the University of Houston. The course is supported in part by NSF Award DMS-2038118. We thank the organizers for the invitation and generous support.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.