Three tutorials explain math computing and machine learning methods
Data, Numbers, and Geometry: Three Tutorials on Numerical Methods, Machine Learning, and Evaluation
Machine Learning
Summary
This work shares three hands-on tutorials designed for applying math and machine learning in research. One tutorial teaches how to calculate using ideas from geometry and calculus with examples on spheres and flat spaces. Another shows how mathematical patterns can shape neural networks, helping check their accuracy and behavior. The last tutorial guides how to measure and present machine learning results clearly with good charts and statistics. The authors provide all resources needed to try these methods yourself.
What this means in practice
- •For scientific computing teams: Use the numerical exterior calculus approach to analyze geometric and topological features from pointwise data in simulations or models.
- •For machine learning engineers: Incorporate mathematical structure into neural network designs to improve learning reliability and validate error bounds over problem domains.
Authors
Jessica N. Howard, Yidi Qi, Tomás S. R. Silva
Abstract
We present three practical tutorials on numerical computation and machine learning for mathematical research, developed for the DANGER: Data, Numbers, and Geometry workshop held at the Banff International Research Station in April 2026. The first develops a numerical approach to exterior calculus from pointwise evaluations of differential forms, using a flux formulation of the exterior derivative. Examples in Euclidean space and on the sphere illustrate geometric identities, topological features, and the effects of approximation and finite precision. The second examines how mathematical structure guides neural network design through examples involving elliptic curves, quivers, and a boundary value problem. It explores how architectural choices affect learning and uses interval arithmetic to bound the residual of a trained network over the full interval of the boundary value problem. The third addresses the evaluation and presentation of machine learning results, covering performance metrics, statistical uncertainty, classification thresholds, receiver operating characteristic curves, and accessible figure design. Throughout, the tutorials distinguish numerical agreement, predictive accuracy, structural guarantees, and rigorous bounds as different forms of evidence. Each contribution can be read independently, with accompanying notebooks and exercises that allow readers to reproduce the examples and adapt the methods to other problems.