Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation

TL;DR

This work demonstrates that a compact analytic model can faithfully reproduce neural surrogates for Calabi–Yau metric determinants governed by the Monge–Ampère PDE. By distilling a five-term formula in terms of gauge-invariant invariants $p_2$ and $\sigma_3$, the authors achieve $R^2\approx0.9994$ with a $3{,}000\times$ reduction in parameters and robust validity across the Dwork family moduli range via volume and Yukawa benchmarks. The functional form remains stable as moduli change, with coefficients $c_i(\psi)$ varying smoothly; singular terms capture essential geometric corrections, revealing a hierarchical modulation that mirrors PDE-constrained structure. The approach significantly accelerates physics calculations by enabling microsecond evaluations, facilitating large-scale moduli scans with practical accuracy limits set by teacher noise. This work also connects with concurrent efforts on symbolic representations of Kahler potentials, highlighting a general principle: fixed PDE structure yields a low-dimensional, interpretable manifold for complex geometric observables.

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 ($ψ\in [0, 0.8]$) with coefficients varying smoothly; we interpret these trends as empirical hypotheses within the accuracy regime of the locally-trained teachers ($σ\approx 8-9\%$ at $ψ\neq 0$). 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.

Figures (7)

• Figure 1: Validation of the symbolic formula (Eq. \ref{['eq:main']}) on $10,000$ independent hold-out test points. (Top) Scatter plot shows near-perfect agreement ($R^2=0.9994$) between symbolic prediction and neural surrogate. (Bottom) Residuals are approximately symmetric with $\sigma \approx 0.011$ and zero mean; formal normality is rejected at $\alpha=0.05$ due to minor tail deviations, but the central 95% follows a near-Gaussian distribution (see Appendix for detailed analysis).
• Figure 2: Coefficient trajectories $c_i(\psi)$ reveal hierarchical moduli response: singular terms ($c_1$, $c_3$) undergo sign reversal (vertical dashed line marks zero crossing), while symmetric term $c_4$ strengthens monotonically. Inset: coefficient classification by response regime.
• Figure 3: Physics Benchmark: Volume integral $V(\psi) = \int_X \det(g) \cdot \omega^3$ computed along a path in moduli space $\psi \in [0, 0.8]$, well outside the training point ($\psi=0$). The symbolic formula (blue dashed), utilizing hierarchically modulated coefficients, tracks the neural surrogate volume (black solid) to within $\approx 2\%$ relative error.
• Figure 4: Cross-k validation: Coefficient variation across polynomial degrees $k = 6, 8, 10$ at $\psi = 0$. Despite different teacher accuracies, the five-term structure remains stable with coefficients varying within $\pm 30\%$ bounds.
• Figure 5: Training curves: Ricci-flatness error $\sigma$ vs. Donaldson iteration for $\psi \in \{0.0, 0.2, 0.4, 0.6, 0.8\}$. All curves converge within 15 iterations.