# Machine Learning Regularization for the Minimum Volume Formula of Toric Calabi-Yau 3-folds

arXiv: https://arxiv.org/abs/2310.19276 · HTML: https://mlsanity.com/paper/2310.19276/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 86 references that are in the map.
- First posted on arXiv 30 Oct 2023.

## Builds on (most cited first, 12 of 12)

- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Neural Networks and Quantum Field Theory](https://mlsanity.com/paper/2008.08601/) (2020)
- [Learning to Unknot](https://mlsanity.com/paper/2010.16263/) (2020)
- [Machine Learning CICY Threefolds](https://mlsanity.com/paper/1806.03121/) (2018)
- [Neural Network Field Theories: Non-Gaussianity, Actions, and Locality](https://mlsanity.com/paper/2307.03223/) (2023)
- [Machine learning for complete intersection Calabi-Yau manifolds: a methodological study](https://mlsanity.com/paper/2007.15706/) (2020)
- [Machine Learning Calabi-Yau Four-folds](https://mlsanity.com/paper/2009.02544/) (2020)
- [Machine Learning Calabi-Yau Hypersurfaces](https://mlsanity.com/paper/2112.06350/) (2021)
- [Neurons on Amoebae](https://mlsanity.com/paper/2106.03695/) (2021)
- [Searching the Landscape of Flux Vacua with Genetic Algorithms](https://mlsanity.com/paper/1907.10072/) (2019)
- [Entangled q-Convolutional Neural Nets](https://mlsanity.com/paper/2103.11785/) (2021)
- [Interpretable Phase Detection and Classification with Persistent Homology](https://mlsanity.com/paper/2012.00783/) (2020)

## Built on by (most cited first, 5 of 5)

- [Learning BPS Spectra and the Gap Conjecture](https://mlsanity.com/paper/2405.09993/) (2024)
- [Generative AI for Brane Configurations and Coamoeba](https://mlsanity.com/paper/2411.16033/) (2024)
- [Learning 3-Manifold Triangulations](https://mlsanity.com/paper/2405.09610/) (2024)
- [Hamiltonian Neural Networks approach to fuzzball geodesics](https://mlsanity.com/paper/2502.20881/) (2025)
- [Calabi-Yau Four/Five/Six-folds as mathbbPnw Hypersurfaces: Machine Learning, Approximation, and Generation](https://mlsanity.com/paper/2311.17146/) (2023)

## Abstract (arXiv)

We present a collection of explicit formulas for the minimum volume of Sasaki-Einstein 5-manifolds. The cone over these 5-manifolds is a toric Calabi-Yau 3-fold. These toric Calabi-Yau 3-folds are associated with an infinite class of 4d N=1 supersymmetric gauge theories, which are realized as worldvolume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under the AdS/CFT correspondence, the minimum volume of the Sasaki-Einstein base is inversely proportional to the central charge of the corresponding 4d N=1 superconformal field theories. The presented formulas for the minimum volume are in terms of geometric invariants of the toric Calabi-Yau 3-folds. These explicit results are derived by implementing machine learning regularization techniques that advance beyond previous applications of machine learning for determining the minimum volume. Moreover, the use of machine learning regularization allows us to present interpretable and explainable formulas for the minimum volume. Our work confirms that, even for extensive sets of toric Calabi-Yau 3-folds, the proposed formulas approximate the minimum volume with remarkable accuracy.