arXiv：cs.LG· Jessica N. Howard, Yidi Qi, Tom\'as S. R. Silva·· 4 小时前AI 评分27
数据、数字与几何：数值方法、机器学习与评估三讲教程
Data, Numbers, and Geometry: Three Tutorials on Numerical Methods, Machine Learning, and Evaluation
AI 导读
一份面向数学研究的教程笔记，源自 2026 年 4 月在 Banff 国际研究站举办的 DANGER: Data, Numbers, and Geometry 研讨会，涵盖三部分内容：从微分形式的逐点求值出发的数值外微积分、数学结构引导神经网络设计的案例（含椭圆曲线、箭图与边值问题）、以及机器学习结果的评估与呈现方法。
正文
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.
| Comments: | Combined notes from three tutorials presented at the DANGER: Data, Numbers, and Geometry workshop (BIRS, Banff, April 2026). Includes links to companion code, Jupyter notebooks, and exercises |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.07220 [cs.LG] |
| (or arXiv:2610.07220v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.07220 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Tomás Silva [view email]
[v1]
Mon, 5 Oct 2026 18:29:40 UTC (2,065 KB)
来源：arXiv：cs.LG · arxiv.org