NSF CBMS 2025: Computational Mathematics and AI
Ten Lectures on Computational Math and AI
Overview
A ten-lecture introductory course on research topics at the intersection of computational mathematics and artificial intelligence. This course exposes how computational mathematics provides foundations, precise language, and design principles for AI, and how AI enables new capabilities for tackling previously intractable computational problems. Topics span machine learning fundamentals, optimization theory and practice, regularization techniques, generative modeling, scientific machine learning, high-dimensional PDEs, inverse problems, and AI-assisted mathematical discovery.
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) Live Q&A: slido.com #CBMS25 Recordings: YouTube Playlist
Lecture Schedule & Topics
| Time | Title | Description | Slides |
|---|---|---|---|
| M 09:00 | Machine Learning Overview | Data-driven approximation framework: bias-variance tradeoff, double descent, and intersection of computational mathematics and AI | |
| M 10:30 | Neural Network Architectures & Loss Functions | How mathematical structures (convolution, attention, depth) encode inductive biases; covers CNNs, GNNs, Transformers, ResNets | |
| T 09:00 | Optimization for Machine Learning | Stochastic vs. sample average approximation, backpropagation, second-order methods, and foundations of SGD | |
| T 10:30 | Why SGD Works | Theory-practice gap: implicit regularization, gradient flow, Langevin dynamics, and benign loss landscape structure | |
| W 09:00 | Adaptive Optimization Methods | Momentum, adaptive methods (Adam, AdamW, Lion), and structured second-order approximations (K-FAC) | |
| W 10:30 | PDE Framework for Generative Modeling | Normalizing flows, optimal transport, flow matching, and diffusion models from classical PDEs | |
| Th 09:00 | Scientific Machine Learning for PDEs | PINNs and neural operators: when do ML methods provide improvements over classical solvers? | |
| Th 10:30 | High-Dimensional PDEs | Breaking curse of dimensionality with neural networks and smart sampling; Hamilton-Jacobi-Bellman equations and mean field games | |
| F 09:00 | Machine Learning for Inverse Problems | Why direct networks fail; Bayesian inference with diffusion priors and posterior sampling | |
| F 10:30 | Mathematical Discovery and Theorem Proving | AI discovers novel algorithms; theorem proving with machine-assisted formal verification in proof assistants |
Lecture Materials
Slides and materials will be posted here as they become available. Please visit the course GitHub repository for code examples.
External Resources
Acknowledgments
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.