AI 中文总结
本教程集面向数学研究，涵盖外微积分数值方法、数学结构引导的神经网络设计及机器学习结果评估，强调区分不同证据形式并提供可复现的实践练习。
AI 中文摘要
我们为数学研究中的数值计算和机器学习提供了三个实用教程，这些教程是为2026年4月在班夫国际研究站举办的DANGER：数据、数字与几何研讨会开发的。第一个教程从微分形式的逐点评估出发，利用外导数的通量公式，发展了一种外微积分的数值方法。欧几里得空间和球面上的例子展示了几何恒等式、拓扑特征以及近似和有限精度的影响。第二个教程通过涉及椭圆曲线、箭图和边值问题的例子，考察数学结构如何指导神经网络设计。它探讨了架构选择如何影响学习，并使用区间算术在边值问题的整个区间上界定训练后网络的残差。第三个教程涉及机器学习结果的评估和呈现，涵盖性能指标、统计不确定性、分类阈值、接收者操作特征曲线以及易读的图形设计。在整个教程中，我们区分了数值一致性、预测准确性、结构保证和严格界限作为不同形式的证据。每个贡献都可以独立阅读，并附有笔记本和练习，使读者能够重现示例并将方法应用于其他问题。
英文摘要
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.
CommentsCombined 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
