Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau $7$-manifolds

TL;DR

This paper demonstrates that machine learning can accurately predict complex topological invariants of contact Calabi-Yau 7-manifolds, significantly speeding up computations and suggesting new mathematical conjectures.

Contribution

It introduces a machine learning framework to predict topological quantities of contact Calabi-Yau 7-manifolds, achieving high accuracy and revealing new conjectures.

Findings

Neural networks achieved R^2 of 0.969 in predicting Sasakian Hodge numbers.

Symbolic regressors achieved R^2 of 0.993, indicating near-perfect predictions.

Machine learning significantly speeds up the computation of topological invariants.

Abstract

We propose a machine learning approach to study topological quantities related to the Sasakian and $G_2$-geometries of contact Calabi-Yau $7$-manifolds. Specifically, we compute datasets for certain Sasakian Hodge numbers and for the Crowley-N\"ordstrom invariant of the natural $G_2$-structure of the $7$-dimensional link of a weighted projective Calabi-Yau $3$-fold hypersurface singularity, for 7549 of the 7555 possible $\mathbb{P}^4(\textbf{w})$ projective spaces. These topological quantities are then machine learnt with high performance scores, where learning the Sasakian Hodge numbers from the $\mathbb{P}^4(\textbf{w})$ weights alone, using both neural networks and a symbolic regressor which achieve $R^2$ scores of 0.969 and 0.993 respectively. Additionally, properties of the respective Gr\"obner bases are well-learnt, leading to a vast improvement in computation speeds which may be of independent interest. The data generation and analysis further induced novel conjectures to be raised.