Numerical Calabi-Yau metrics from holomorphic networks

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 ...

We introduce neural networks to compute numerical Ricci-flat CY metrics for complete intersection and Kreuzer-Skarke Calabi-Yau manifolds at any point in K\"ahler and complex structure moduli space, and introduce the package cymetric which provides computation realizations of these techniques. In particular, we develop and computationally realize methods for point-sampling on these manifolds. The training for the neural networks is carried out subject to a custom loss functio...

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...

We present a new machine learning library for computing metrics of string compactification spaces. We benchmark the performance on Monte-Carlo sampled integrals against previous numerical approximations and find that our neural networks are more sample- and computation-efficient. We are the first to provide the possibility to compute these metrics for arbitrary, user-specified shape and size parameters of the compact space and observe a linear relation between optimization of...

We apply machine learning to the problem of finding numerical Calabi-Yau metrics. We extend previous work on learning approximate Ricci-flat metrics calculated using Donaldson's algorithm to the much more accurate "optimal" metrics of Headrick and Nassar. We show that machine learning is able to predict the K\"ahler potential of a Calabi-Yau metric having seen only a small sample of training data.

Calabi-Yau (CY) manifolds play a ubiquitous role in string theory. As a supersymmetry-preserving choice for the 6 extra compact dimensions of superstring compactifications, these spaces provide an arena in which to explore the rich interplay between physics and geometry. These lectures will focus on compact CY manifolds and the long standing problem of determining their Ricci flat metrics. Despite powerful existence theorems, no analytic expressions for these metrics are know...

We develop a theory of flows in the space of Riemannian metrics induced by neural network gradient descent. This is motivated in part by recent advances in approximating Calabi-Yau metrics with neural networks and is enabled by recent advances in understanding flows in the space of neural networks. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel, a complicated, non-local object that evolves in time. However, many archite...

Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric within a given K\"ahler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi-Yau threefolds. Using these Ricci-flat metric approximations for the Cefal\'u family of quartic twofol...

We present new invariant machine learning models that approximate the Ricci-flat metric on Calabi-Yau (CY) manifolds with discrete symmetries. We accomplish this by combining the $\phi$-model of the cymetric package with non-trainable, $G$-invariant, canonicalization layers that project the $\phi$-model's input data (i.e. points sampled from the CY geometry) to the fundamental domain of a given symmetry group $G$. These $G$-invariant layers are easy to concatenate, provided o...

We review advancements in deep learning techniques for complete intersection Calabi-Yau (CICY) 3- and 4-folds, with the aim of understanding better how to handle algebraic topological data with machine learning. We first discuss methodological aspects and data analysis, before describing neural networks architectures. Then, we describe the state-of-the art accuracy in predicting Hodge numbers. We include new results on extrapolating predictions from low to high Hodge numbers,...