Yau's theorem guarantees the existence of a unique Ricci-flat K ahler metric with a given K ahler class on a Calabi-Yau (CY) manifold. Such Ricci-flat metrics are of mathematical interest and they play an important role in compactifications of string theory. Unfortunately, for compact Calabi-Yau manifolds of complex dimension three or higher, analytic expressions for Ricci-flat metrics are not known. Over the past few years substantial progress has nevertheless been made in numerically computing Ricci-flat metrics on Calabi-Yau three-folds, starting with Donaldson's algorithm [1] and its applications [2-9] and, more recently, using machine learning methods [10-16]. The Ricci-flat metric in numerical form is already useful, enabling us to compute the spectrum of the Laplacian on a CY manifold [5,17] or the masses of quarks in a CY string compactification [18], to name just two applications. However, it would be all the more helpful and exciting to deal with the metric analytically.

We apply machine learning to the problem of finding numerical Calabi-Yau metrics. Building on Donaldson's algorithm for calculating balanced metrics on K\"ahler manifolds, we combine conventional curve fitting and machine-learning techniques to numerically approximate Ricci-flat metrics. We show that machine learning is able to predict the Calabi-Yau metric and quantities associated with it, such as its determinant, having seen only a small sample of training data. Using this in conjunction with a straightforward curve fitting routine, we demonstrate that it is possible to find highly accurate numerical metrics much more quickly than by using Donaldson's algorithm alone, with our new machine-learning algorithm decreasing the time required by between one and two orders of magnitude.