[Paper Review] Numerical Calabi-Yau metrics from holomorphic networks

Michael R. Douglas, Subramanian Lakshminarasimhan+1 more|arXiv (Cornell University)|Dec 9, 2020

Geometry and complex manifolds44 references22 citations

TL;DR

This paper introduces a machine learning-inspired method to compute numerical Calabi-Yau metrics using holomorphic and bihomogeneous neural networks in TensorFlow/Keras. It achieves ~0.1% mean absolute percentage error (MAPE) on asymmetric quintic threefolds—100× more accurate than prior methods—by representing the Kähler potential via deep networks and optimizing for Ricci flatness via gradient descent.

ABSTRACT

We propose machine learning inspired methods for computing numerical Calabi-Yau (Ricci flat Kähler) metrics, and implement them using Tensorflow/Keras. We compare them with previous work, and find that they are far more accurate for manifolds with little or no symmetry. We also discuss issues such as overparameterization and choice of optimization methods.

Motivation & Objective

• To develop a scalable, high-accuracy numerical method for computing Ricci-flat Kähler metrics on Calabi-Yau manifolds, especially those with low or no symmetry.
• To overcome the limitations of traditional spectral methods, which suffer from the curse of dimensionality and require high symmetry to remain computationally feasible.
• To explore the use of deep feedforward neural networks—specifically holomorphic and bihomogeneous networks—as a parameterization of the Kähler potential for solving high-dimensional nonlinear PDEs.
• To evaluate the performance of these methods in terms of accuracy, optimization stability, and generalization across diverse Calabi-Yau geometries.
• To understand the role of model capacity, depth, width, and training data size in achieving accurate Ricci-flat metric approximations.

Proposed method

• Represent the Kähler potential as the output of a bihomogeneous feedforward neural network, where inputs are real and imaginary parts of holomorphic and antiholomorphic monomials, and weights are real-valued.
• Use a holomorphic network with complex weights and activations to generate holomorphic sections, then form a Hermitian combination to produce a real Kähler potential.
• Optimize the network weights via gradient descent to minimize a loss function measuring Ricci flatness, defined as the L2 norm of the Ricci curvature.
• Train using a set of sampled points on the Calabi-Yau manifold, with the loss function evaluated at these points to approximate the global L2 error.
• Employ a two-stage optimization: first use Adam optimizer for convergence, then switch to L-BFGS for higher precision, limited by GPU memory to ~2000 points.
• Use a benchmarking framework with test cases and a public leaderboard to compare results across different network architectures and hyperparameters.

Experimental results

Research questions

• RQ1 Can deep neural networks with complex or real weights effectively parameterize the Kähler potential to achieve high-accuracy Ricci-flat metrics on asymmetric Calabi-Yau manifolds?
• RQ2 How does the accuracy of the metric approximation depend on network depth, width, and the number of training points?
• RQ3 What is the role of model overparameterization in achieving low training and generalization error for Calabi-Yau metric computation?
• RQ4 How do the bihomogeneous and holomorphic network architectures compare in terms of optimization stability and final accuracy?
• RQ5 What is the theoretical limit of accuracy achievable for a given network depth and parameter count, and how does it scale with the complexity of the underlying Calabi-Yau geometry?

Key findings

• The method achieves a mean absolute percentage error (MAPE) of approximately 0.1% on Calabi-Yau threefolds with no symmetry, representing a ~100-fold improvement over previous methods like Braun et al. (2008a).
• The bihomogeneous network architecture outperforms the holomorphic network due to better optimization stability and reduced susceptibility to bad local minima.
• Four-layer networks with widths around 100 (e.g., 100_100_100_1) provide the best balance of accuracy and reliability, outperforming deeper or wider configurations like 500_500_500_500_1.
• Optimization with L-BFGS after Adam pre-training improves accuracy, but memory constraints limit training to ~2000 points, reducing reliability of high-accuracy runs.
• Overparameterized models (with $ N_p imes P $) can achieve zero training loss for $ N_p \leq 2500 $, but testing error scales as $ 1/\sqrt{N_p} $, indicating generalization is sensitive to data size.
• Random label experiments confirm the model's ability to memorize data, supporting the interpretation that generalization is governed by inductive bias and model capacity rather than overfitting alone.