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 and 97% for (100% for both), 81% (96%) for , and 49% (83%) for . 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.
Highlights
Computing Hodge numbers of Calabi-Yau manifolds is of great importance, since cohomology computations are an integral part of string theory compactifications
The model we developed is capable of learning at the same time, and without rescaling, the four dimensions of the tangent space cohomologies of CICYs, accounting for the heavy class imbalance present in the dataset
We were able to show that Inception-based neural networks achieve good accuracy in predicting h(3,1) and h(2,2) and can reach perfect accuracy for the Hodge numbers h(1,1), h(2,1)
Summary
The first paper utilizing machine learning algorithm to predict various different cohomology dimensions was written by He [20]. In line with previous studies of h(2,1) on CICY three-folds, the authors were unable to accurately predict the value of the other Hodge numbers, reaching an accuracy of only 27 % for h(3,1). They were successful in improving this accuracy for a subset of the dataset by considering all configuration matrices of shape (4, 4) and using feature enhancement. In order to identify equivalent Calabi-Yau manifolds coming from different triangulations, Demirtas et al trained residual neural networks to learn the triple intersection numbers [36] They reached an almost perfect performance, which allowed them to cut down the computation time from seconds to microseconds. The Hodge numbers belonging to these cones can all be described by the same analytic equations [43]
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