# Learning Size and Shape of Calabi-Yau Spaces

arXiv: https://arxiv.org/abs/2111.01436 · HTML: https://mlsanity.com/paper/2111.01436/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 43 references that are in the map.
- First posted on arXiv 2 Nov 2021.

## Builds on (most cited first, 4 of 4)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Neural Network Approximations for Calabi-Yau Metrics](https://mlsanity.com/paper/2012.15821/) (2020)
- [Machine learning Calabi-Yau metrics](https://mlsanity.com/paper/1910.08605/) (2019)
- [Unsupervised Neural Networks for Quantum Eigenvalue Problems](https://mlsanity.com/paper/2010.05075/) (2020)

## Built on by (most cited first, 28 of 28)

- [Machine Learning Calabi-Yau Hypersurfaces](https://mlsanity.com/paper/2112.06350/) (2021)
- [Machine Learned Calabi-Yau Metrics and Curvature](https://mlsanity.com/paper/2211.09801/) (2022)
- [Rigor with Machine Learning from Field Theory to the Poincar'e Conjecture](https://mlsanity.com/paper/2402.13321/) (2024)
- [Calabi-Yau Metrics, Energy Functionals and Machine-Learning](https://mlsanity.com/paper/2112.10872/) (2021)
- [Metric Flows with Neural Networks](https://mlsanity.com/paper/2310.19870/) (2023)
- [Machine Learning Kreuzer--Skarke Calabi--Yau Threefolds](https://mlsanity.com/paper/2112.09117/) (2021)
- [Physics-informed neural network solves minimal surfaces in curved spacetime](https://mlsanity.com/paper/2509.10866/) (2025)
- [A Machine Learning Approach to the Nirenberg Problem](https://mlsanity.com/paper/2602.12368/) (2026)
- [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)
- [Machine Learning Algebraic Geometry for Physics](https://mlsanity.com/paper/2204.10334/) (2022)
- [cymyc -- Calabi-Yau Metrics, Yukawas, and Curvature](https://mlsanity.com/paper/2410.19728/) (2024)
- [From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook](https://mlsanity.com/paper/2202.06086/) (2022)
- [Generative AI for Brane Configurations and Coamoeba](https://mlsanity.com/paper/2411.16033/) (2024)
- [Minimal surfaces, Knots, and Neural Networks](https://mlsanity.com/paper/2605.26234/) (2026)
- [Intelligent Explorations of the String Theory Landscape](https://mlsanity.com/paper/2204.08073/) (2022)
- [Machine Learning on generalized Complete Intersection Calabi-Yau Manifolds](https://mlsanity.com/paper/2209.10157/) (2022)
- [Neural Network Learning and Quantum Gravity](https://mlsanity.com/paper/2403.03245/) (2024)
- [Hamiltonian Neural Networks approach to fuzzball geodesics](https://mlsanity.com/paper/2502.20881/) (2025)
- [Approximate Ricci-flat Metrics for Calabi-Yau Manifolds](https://mlsanity.com/paper/2506.15766/) (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)
- [Characterizing 4-string contact interaction using machine learning](https://mlsanity.com/paper/2211.09129/) (2022)
- [Calabi-Yau Four/Five/Six-folds as mathbbPnw Hypersurfaces: Machine Learning, Approximation, and Generation](https://mlsanity.com/paper/2311.17146/) (2023)
- [Learning to be Simple](https://mlsanity.com/paper/2312.05299/) (2023)
- [Machine Learning the 6d Supergravity Landscape](https://mlsanity.com/paper/2505.16131/) (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)

We present a new machine learning library for computing metrics of string compactification spaces. We benchmark the performance on Monte-Carlo sampled integrals against previous numerical approximations and find that our neural networks are more sample- and computation-efficient. We are the first to provide the possibility to compute these metrics for arbitrary, user-specified shape and size parameters of the compact space and observe a linear relation between optimization of the partial differential equation we are training against and vanishing Ricci curvature.