Neural and numerical methods for $\mathrm{G}_2$-structures on contact Calabi-Yau 7-manifolds

TL;DR

This paper develops a numerical framework combining neural networks and geometric analysis to approximate $ ext{G}_2$-structures on contact Calabi-Yau 7-manifolds, enabling the study of their geometric properties.

Contribution

It introduces a novel three-stage approach that integrates neural network models with geometric constructions to approximate $ ext{G}_2$-structures and their metrics on contact Calabi-Yau manifolds.

Findings

Successfully approximated Ricci-flat metrics on Calabi-Yau threefolds.

Generated numerical $ ext{G}_2$-forms on 7-manifolds from sampled data.

Validated the learned structures through numerical exterior derivative computations.

Abstract

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.