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We propose a machine-learning approach to study topological quantitiesrelated to the Sasakian and $G_2$-geometries of contact Calabi-Yau$7$-manifolds. Specifically, we compute datasets for certain Sasakian Hodgenumbers 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 each of the 7555 possible$\mathbb{P}^4(\textbf{w})$ projective spaces. These topological quantities arethen machine learnt with high accuracy, along with properties of the respectiveGr\"obner basis, leading to a vast improvement in computation speeds which maybe of independent interest. We observe promising results in machine learningthe Sasakian Hodge numbers from the $\mathbb{P}^4(\textbf{w})$ weights alone,using both neural networks and a symbolic regressor which achieve $R^2$ scoresof 0.969 and 0.993 respectively, inducing novel conjectures to be raised.

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