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

Motivated by recent progress in the problem of numerical Kähler 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 Kähler 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.

💡 Research Summary

The paper provides a comprehensive review of recent advances in the numerical computation of Calabi–Yau (CY) metrics, focusing on the intersection of traditional algebraic‑geometric methods and modern machine‑learning (ML) techniques. After a historical overview that connects Calabi’s conjecture, Yau’s proof, and the role of Ricci‑flat Kähler metrics in string‑theoretic compactifications, the authors discuss the limitations of early numerical approaches such as the Headrick‑Wiseman Kähler potential discretization and Donaldson’s algorithm based on global holomorphic sections of an ample line bundle. While Donaldson’s method offers a mathematically rigorous convergence guarantee for constant‑scalar‑curvature Kähler (cscK) metrics, it suffers from the “curse of dimensionality” because the number of sections grows rapidly with the line‑bundle degree.

The paper then surveys recent ML‑based attempts to approximate CY metrics, including physics‑informed neural networks (PINNs) and direct neural‑network parametrizations of the metric. The authors acknowledge the practical advantages of these approaches—parallel GPU execution, automatic differentiation, and flexible loss design—but also highlight serious drawbacks: loss functions may not enforce positivity of the metric, the black‑box nature hampers interpretability, and error accumulation can lead to non‑Kähler or non‑Ricci‑flat outputs.

Motivated by these observations, the authors propose a hybrid framework that retains the algebraic structure of Donaldson’s ansatz while embedding it in a differentiable optimization landscape. The key idea is to treat the space of candidate sections as points on a Grassmannian manifold $\mathrm{Gr}(k,N)$, where $N$ is the total number of global sections at a given degree and $k$ is the number of sections actually used in the metric construction. By endowing the Grassmannian with its natural Riemannian metric, they perform gradient descent directly on this manifold to discover an optimal low‑dimensional subspace of sections. This “Grassmannian learning” dramatically reduces the computational burden because only a small, well‑chosen basis is needed to achieve a given accuracy.

In parallel, the authors treat the Hermitian matrix $h$ (which encodes the inner products of the chosen sections) as a learnable variable living in the fiber bundle of Hermitian metrics over the Grassmannian. Gradient descent on $h$ is equivalent to performing a continuous version of Donaldson’s iterative “$T$‑operator” but benefits from modern stochastic optimization techniques, adaptive learning rates, and back‑propagation. Importantly, the loss function explicitly includes the Kähler condition $d\omega=0$ and a positivity penalty, guaranteeing that the resulting metric remains Kähler and positive‑definite throughout training.

The methodology is tested on the one‑parameter Dwork family of quintic threefolds defined by \