Learning Group Invariant Calabi-Yau Metrics by Fundamental Domain Projections

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 $\mathbb{Z}<sup>2_5$ quotient of a 5-parameter quintic CY manifold.