AInstein: Numerical Einstein Metrics via Machine Learning
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909853635575808 |
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| author | Hirst, Edward Gherardini, Tancredi Schettini Stapleton, Alexander G. |
| author_facet | Hirst, Edward Gherardini, Tancredi Schettini Stapleton, Alexander G. |
| contents | 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 $R_{μν} - λg_{μν} = 0$ being used as independent loss components (here $μ,ν= 1, 2, \cdots, n$, where $n$ is the dimension of the Riemannian manifold, and the Einstein constant $λ\in \{+1, 0, -1\}$). 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, $g' = J^T g J$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13043 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | AInstein: Numerical Einstein Metrics via Machine Learning Hirst, Edward Gherardini, Tancredi Schettini Stapleton, Alexander G. High Energy Physics - Theory General Relativity and Quantum Cosmology Differential Geometry 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 $R_{μν} - λg_{μν} = 0$ being used as independent loss components (here $μ,ν= 1, 2, \cdots, n$, where $n$ is the dimension of the Riemannian manifold, and the Einstein constant $λ\in \{+1, 0, -1\}$). 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, $g' = J^T g J$, 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. |
| title | AInstein: Numerical Einstein Metrics via Machine Learning |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Differential Geometry |
| url | https://arxiv.org/abs/2502.13043 |