Neural Network Approximations for Calabi-Yau Metrics
This work provides a numerical method for approximating Ricci flat metrics, which are analytically unknown, for mathematicians and theoretical physicists studying Calabi-Yau manifolds.
This paper uses a single neural network architecture to approximate Ricci flat Kaehler metrics for several Calabi-Yau manifolds, including the Fermat quintic, Dwork quintic, and Tian-Yau manifold. After training, measures assessing Ricci flatness decreased by three orders of magnitude on the training set.
Ricci flat metrics for Calabi-Yau threefolds are not known analytically. In this work, we employ techniques from machine learning to deduce numerical flat metrics for the Fermat quintic, for the Dwork quintic, and for the Tian-Yau manifold. This investigation employs a single neural network architecture that is capable of approximating Ricci flat Kaehler metrics for several Calabi-Yau manifolds of dimensions two and three. We show that measures that assess the Ricci flatness of the geometry decrease after training by three orders of magnitude. This is corroborated on the validation set, where the improvement is more modest. Finally, we demonstrate that discrete symmetries of manifolds can be learned in the process of learning the metric.