用神经网络逼近接触卡拉比-丘7流形上的G2结构3形式。
Neural and numerical methods for $\mathrm{G}_2$-structures on contact Calabi-Yau 7-manifolds
- 先用神经网络算出卡拉比-丘三流形的近似里奇平坦度量。
- 再基于此度量和显式构造，数值生成大量点上的3形式近似值。
- 训练专用神经网络直接学习3形式与对应度量，验证其外微分结果。
本文提出一种数值框架，用于逼近接触卡拉比-丘7流形上的G2结构3形式。流程分三步：首先，利用现有神经网络模型计算卡拉比-丘三流形上的近似里奇平坦度量；其次，结合该度量与在9维球面上的显式构造，对大量采样点生成G2结构3形式的数值近似；最后，设计专用神经架构，直接从数据中学习3形式及其诱导的黎曼度量，并通过数值实现外微分来验证所学结构及其挠率，该验证方法本身可能具有独立价值。
原文摘要 · Abstract (English)
A numerical framework for approximating $\mathrm{G}_2$-structure 3-forms on contact Calabi-Yau manifolds is presented. The approach proceeds in three stages: first, existing neural network models are employed to compute an approximate Ricci-flat metric on a Calabi-Yau threefold. Second, using this metric and the explicit construction of a $\mathrm{G}_2$-structure on the associated 7-dimensional Calabi-Yau link in the 9-sphere, numerical approximations of the 3-form are generated on a large set of sampled points. Finally, a dedicated neural architecture is trained to learn the 3-form and its induced Riemannian metric directly from data, validating the learned structure and its torsion via a numerical implementation of the exterior derivative, which may be of independent interest.
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