Deep multi-task mining Calabi-Yau four-folds

TL;DR

This paper applies advanced deep learning techniques to accurately predict all four non-trivial Hodge numbers of Calabi-Yau four-folds, significantly improving previous benchmarks and demonstrating the effectiveness of multi-task neural networks in complex geometric computations.

Contribution

It introduces a multi-task neural network architecture that simultaneously learns all four Hodge numbers of Calabi-Yau four-folds, achieving near-perfect accuracy and advancing computational methods in algebraic geometry.

Findings

Achieved 100% accuracy for $h^{(1,1)}$ and $h^{(2,1)}$ with 30% training data.

Achieved 81% accuracy for $h^{(3,1)}$ and 49% for $h^{(2,2)}$ with 30% training data.

Using the Euler number constraint yields 100% total accuracy.

Abstract

We continue earlier efforts in computing the dimensions of tangent space cohomologies of Calabi-Yau manifolds using deep learning. In this paper, we consider the dataset of all Calabi-Yau four-folds constructed as complete intersections in products of projective spaces. Employing neural networks inspired by state-of-the-art computer vision architectures, we improve earlier benchmarks and demonstrate that all four non-trivial Hodge numbers can be learned at the same time using a multi-task architecture. With 30% (80%) training ratio, we reach an accuracy of 100% for $h^{(1,1)}$ and 97% for $h^{(2,1)}$ (100% for both), 81% (96%) for $h^{(3,1)}$, and 49% (83%) for $h^{(2,2)}$. Assuming that the Euler number is known, as it is easy to compute, and taking into account the linear constraint arising from index computations, we get 100% total accuracy.