Learning group invariant Calabi–Yau metrics by fundamental domain projections

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 state of the art models for predicting such metrics, based on the so-called φ-model of the cymetric package, with non-trainable, G-invariant, canonicalization layers that project the φ-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 both standard and spectral versions of the φ-model. 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 φ-model. On highly symmetric spaces, we also observe significantly faster convergence upon training. The method may also be used to compute the Ricci-flat metric on smooth CY quotients. We demonstrate this aspect by experiments on a smooth Z52 quotient of a 5-parameter quintic CY manifold.