# Machine Learned Calabi-Yau Metrics and Curvature

arXiv: https://arxiv.org/abs/2211.09801 · HTML: https://mlsanity.com/paper/2211.09801/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 61 references that are in the map.
- First posted on arXiv 17 Nov 2022.

## Builds on (most cited first, 3 of 3)

- [Neural Network Approximations for Calabi-Yau Metrics](https://mlsanity.com/paper/2012.15821/) (2020)
- [Learning Size and Shape of Calabi-Yau Spaces](https://mlsanity.com/paper/2111.01436/) (2021)
- [Machine learning Calabi-Yau metrics](https://mlsanity.com/paper/1910.08605/) (2019)

## Built on by (most cited first, 14 of 14)

- [A Machine Learning Approach to the Nirenberg Problem](https://mlsanity.com/paper/2602.12368/) (2026)
- [Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm](https://mlsanity.com/paper/2410.11284/) (2024)
- [Symbolic Approximations to Ricci-flat Metrics Via Extrinsic Symmetries of Calabi-Yau Hypersurfaces](https://mlsanity.com/paper/2412.19778/) (2024)
- [Neural and numerical methods for G2-structures on contact Calabi-Yau 7-manifolds](https://mlsanity.com/paper/2602.12438/) (2026)
- [Minimising Willmore Energy via Neural Flow](https://mlsanity.com/paper/2604.04321/) (2026)
- [cymyc -- Calabi-Yau Metrics, Yukawas, and Curvature](https://mlsanity.com/paper/2410.19728/) (2024)
- [Generative AI for Brane Configurations and Coamoeba](https://mlsanity.com/paper/2411.16033/) (2024)
- [Hamiltonian Neural Networks approach to fuzzball geodesics](https://mlsanity.com/paper/2502.20881/) (2025)
- [Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation](https://mlsanity.com/paper/2602.07834/) (2026)
- [PINNs in More General Geometry](https://mlsanity.com/paper/2604.25020/) (2026)
- [Black holes and the loss landscape in machine learning](https://mlsanity.com/paper/2306.14817/) (2023)
- [Calabi-Yau Four/Five/Six-folds as mathbbPnw Hypersurfaces: Machine Learning, Approximation, and Generation](https://mlsanity.com/paper/2311.17146/) (2023)
- [Hermitian Yang--Mills connections on general vector bundles: geometry and physical Yukawa couplings](https://mlsanity.com/paper/2512.10907/) (2025)
- [GlobalCY I: A JAX Framework for Globally Defined and Symmetry-Aware Neural K"ahler Potentials](https://mlsanity.com/paper/2604.11404/) (2026)

## Abstract (arXiv)

Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric within a given K"ahler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi-Yau threefolds. Using these Ricci-flat metric approximations for the Cefal'u family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely Spectral networks, which correctly approximate the topological characteristic of a Calabi-Yau. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points. For our neural network approximations, we observe a Bogomolov--Yau type inequality 3c2 ≥ c12 and observe an identity when our geometries have isolated A1 type singularities. We sketch a proof that chi(X~smallsetminus~Sing,X) + 2~|Sing,X| = 24 also holds for our numerical approximations.