Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation

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.