Learning Group Invariant Calabi-Yau Metrics by Fundamental Domain Projections

TL;DR

This paper introduces invariant machine learning models that approximate Ricci-flat metrics on Calabi-Yau manifolds with symmetries by projecting data onto fundamental domains, improving accuracy over standard models.

Contribution

It combines the $ ext{cymetric}$ package with $G$-invariant canonicalization layers to enhance metric approximation on symmetric Calabi-Yau manifolds.

Findings

Canonicalized models outperform standard $ ext{phi}$-models in accuracy.

The method effectively computes Ricci-flat metrics on quotient CY manifolds.

Models are easily concatenated and compatible with spectral $ ext{phi}$-models.

Abstract

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 one compatibility condition is fulfilled, and combine well with spectral $\phi$-models. Through experiments on different CY geometries, we find that, for fixed point sample size and training time, canonicalized models give slightly more accurate metric approximations than the standard $\phi$-model. The method may also be used to compute Ricci-flat metric on smooth CY quotients. We demonstrate this aspect by experiments on a smooth $\mathbb{Z}^2_5$ quotient of a 5-parameter quintic CY manifold.