Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation

TL;DR

This paper demonstrates that symbolic regression can distill neural network approximations of Calabi-Yau metrics into simple, interpretable formulas that maintain high accuracy and physical relevance across a range of parameters.

Contribution

It introduces a method to extract compact, interpretable formulas from neural models of Calabi-Yau metrics, enhancing understanding and computational efficiency.

Findings

Symbolic formulas match neural accuracy with far fewer parameters.

Geometric constraints highlight essential features like power sums and symmetric polynomials.

The formulas accurately reproduce physical observables such as volume integrals and Yukawa couplings.

Abstract

Calabi--Yau manifolds are essential for string theory but require computing intractable metrics. Here we show that symbolic regression can distill neural approximations into simple, interpretable formulas. Our five-term expression matches neural accuracy ($R^2 = 0.9994$) with 3,000-fold fewer parameters. Multi-seed validation confirms that geometric constraints select essential features, specifically power sums and symmetric polynomials, while permitting structural diversity. The functional form can be maintained across the studied moduli range ($\psi \in [0, 0.8]$) with coefficients varying smoothly; we interpret these trends as empirical hypotheses within the accuracy regime of the locally-trained teachers ($\sigma \approx 8-9\%$ at $\psi \neq 0$). The formula reproduces physical observables -- volume integrals and Yukawa couplings -- validating that symbolic distillation recovers compact, interpretable models for quantities previously accessible only to black-box networks.