Neural network approximations for Calabi-Yau metrics

Abstract

Abstract 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 K3, the Fermat quintic, and the Dwork quintic. This investigation employs a simple, modular neural network architecture that is capable of approximating Ricci flat Kähler metrics for Calabi-Yau manifolds of dimensions two and three. We show that measures that assess the Ricci flatness and consistency of the metric decrease after training. This improvement is corroborated by the performance of the trained network on an independent validation set. Finally, we demonstrate the consistency of the learnt metric by showing that it is invariant under the discrete symmetries it is expected to possess.