February 18, 2025
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 that patch, with the associated Einstein conditions, of the form $R_{\mu \nu} - \lambda g_{\mu \nu} = 0$, being used as independent loss components; in our conventions, $\mu,\nu = 1, 2, \cdots, n$, where $n$ is the dimension of the Riemannian manifold and $\lambda \in \{+1, 0, -1\}$. 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' = J g J^T$, for an appropriate analytically known Jacobian $J$. We test our method for the case of spheres represented with 2 patches in dimensions $2,3,4,5$; in dimensions $2, 3$ the geometries have been fully classified, however it is unknown whether a Ricci-flat metric can be put on spheres in dimensions $4, 5$, which we provide numerical evidence against.
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