# Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm

arXiv: https://arxiv.org/abs/2410.11284 · HTML: https://mlsanity.com/paper/2410.11284/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 81 references that are in the map.
- First posted on arXiv 15 Oct 2024.

## Builds on (most cited first, 15 of 15)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Learning Optimal Solutions for Extremely Fast AC Optimal Power Flow](https://mlsanity.com/paper/1910.01213/) (2019)
- [The loss surface of deep and wide neural networks](https://mlsanity.com/paper/1704.08045/) (2017)
- [Imposing Hard Constraints on Deep Networks: Promises and Limitations](https://mlsanity.com/paper/1706.02025/) (2017)
- [Neural Network Approximations for Calabi-Yau Metrics](https://mlsanity.com/paper/2012.15821/) (2020)
- [Achieving Conservation of Energy in Neural Network Emulators for Climate Modeling](https://mlsanity.com/paper/1906.06622/) (2019)
- [Exact Solutions of a Deep Linear Network](https://mlsanity.com/paper/2202.04777/) (2022)
- [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, Energy Functionals and Machine-Learning](https://mlsanity.com/paper/2112.10872/) (2021)
- [How regularization affects the critical points in linear networks](https://mlsanity.com/paper/1709.09625/) (2017)

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

- [Symbolic Approximations to Ricci-flat Metrics Via Extrinsic Symmetries of Calabi-Yau Hypersurfaces](https://mlsanity.com/paper/2412.19778/) (2024)
- [Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation](https://mlsanity.com/paper/2602.07834/) (2026)
- [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)

Motivated by recent progress in the problem of numerical K"ahler metrics, we survey machine learning techniques in this area, discussing both advantages and drawbacks. We then revisit the algebraic ansatz pioneered by Donaldson. Inspired by his work, we present a novel approach to obtaining Ricci-flat approximations to K"ahler metrics, applying machine learning within a `principled' framework. In particular, we use gradient descent on the Grassmannian manifold to identify an efficient subspace of sections for calculation of the metric. We combine this approach with both Donaldson's algorithm and learning on the h-matrix itself (the latter method being equivalent to gradient descent on the fibre bundle of Hermitian metrics on the tautological bundle over the Grassmannian). We implement our methods on the Dwork family of threefolds, commenting on the behaviour at different points in moduli space. In particular, we observe the emergence of nontrivial local minima as the moduli parameter is increased.