Neural and numerical methods for $\mathrm{G}_2$-structures on contact Calabi-Yau 7-manifolds
This work addresses a specific challenge in differential geometry and mathematical physics related to G2-structures, but it is incremental as it builds on existing neural network models and numerical techniques.
The paper tackles the problem of approximating G2-structure 3-forms on contact Calabi-Yau 7-manifolds by developing a numerical framework that combines neural networks and explicit constructions, resulting in a method to learn the 3-form and its induced metric from sampled data.
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.