AInstein: numerical Einstein metrics via machine learning

Abstract A new semi-supervised machine learning package is introduced which successfully solves the Euclidean vacuum Einstein equations with a cosmological constant, without any symmetry assumptions. The model architecture contains subnetworks for each patch in the manifold-defining atlas. Each subnetwork predicts the components of a metric in its associated patch, with the relevant Einstein conditions of the form $R_{\mu \nu }-\lambda g_{\mu \nu }=0$ being used as independent loss components (here $\mu ,\nu =1,2,\cdots ,n$, where n is the dimension of the Riemannian manifold, and the Einstein constant $\lambda \in +1,0,-1$). To ensure the consistency of the global structure of the manifold, another loss component is introduced across the patch subnetworks which enforces the coordinate transformation between the patches, $g^{\prime }=J^{T}gJ$, for an appropriate analytically known Jacobian J. We test our method for the case of spheres represented by a pair of patches in dimensions 2, 3, 4, and 5. In dimensions 2 and 3, the geometries have been fully classified. However, it is unknown whether a Ricci-flat metric can exist on spheres in dimensions 4 and 5. This work hints against the existence of such a metric.