# Symbolic Approximations to Ricci-flat Metrics Via Extrinsic Symmetries of Calabi-Yau Hypersurfaces

arXiv: https://arxiv.org/abs/2412.19778 · HTML: https://mlsanity.com/paper/2412.19778/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 64 references that are in the map.
- First posted on arXiv 27 Dec 2024.

## Builds on (most cited first, 13 of 13)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [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)
- [Machine Learned Calabi-Yau Metrics and Curvature](https://mlsanity.com/paper/2211.09801/) (2022)
- [Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm](https://mlsanity.com/paper/2410.11284/) (2024)
- [cymyc -- Calabi-Yau Metrics, Yukawas, and Curvature](https://mlsanity.com/paper/2410.19728/) (2024)

## Built on by (most cited first, 4 of 4)

- [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)
- [GlobalCY I: A JAX Framework for Globally Defined and Symmetry-Aware Neural K"ahler Potentials](https://mlsanity.com/paper/2604.11404/) (2026)

## Abstract (arXiv)

Ever since Yau's non-constructive existence proof of Ricci-flat metrics on Calabi-Yau manifolds, finding their explicit construction remains a major obstacle to development of both string theory and algebraic geometry. Recent computational approaches employ machine learning to create novel neural representations for approximating these metrics, offering high accuracy but limited interpretability. In this paper, we analyse machine learning approximations to flat metrics of Fermat Calabi-Yau n-folds and some of their one-parameter deformations in three dimensions in order to discover their new properties. We formalise cases in which the flat metric has more symmetries than the underlying manifold, and prove that these symmetries imply that the flat metric admits a surprisingly compact representation for certain choices of complex structure moduli. We show that such symmetries uniquely determine the flat metric on certain loci, for which we present an analytic form. We also incorporate our theoretical results into neural networks to reduce Ricci curvature for multiple Calabi--Yau manifolds compared to previous machine learning approaches. We conclude by distilling the ML models to obtain for the first time closed form expressions for Kahler metrics with near-zero scalar curvature.