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

2026-02-12

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• G2-structure
• Calabi-Yau manifold
• numerical approximation
• differential forms
• Ricci-flat metric

Innovation

• G2-structure
• neural networks
• Calabi-Yau manifolds
• numerical approximation
• exterior derivative