Data, Numbers, and Geometry: Three Tutorials on Numerical Methods, Machine Learning, and Evaluation

2026-10-05

AI Summary

This project addresses the practical challenges of integrating numerical computation with machine learning in mathematical research by proposing a systematic solution. Methodologically, it combines exterior differential flux formulas, structure-guided neural network design, interval arithmetic residual bounding, and statistical evaluation techniques to explicitly distinguish three forms of evidence: numerical consistency, predictive accuracy, and rigorous bounds. The primary contributions comprise three independently readable practical tutorials accompanied by computational notebooks, which effectively support example reproduction and methodological transfer. By providing standardized guidance for engineering implementation, this work facilitates the adaptation of these integrated approaches across related domains.

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.

Problem

• numerical methods
• machine learning
• exterior calculus
• neural network design
• evaluation metrics

Innovation

• Exterior Calculus
• Interval Arithmetic
• Neural Network Architecture
• Mathematical Structure
• Rigorous Bounds