Calabi-Yau Metrics with Full Moduli Dependence

Recent advances in numerical and machine-learning methods have enabled highly accurate constructions of Ricci-flat metrics on compact Calabi-Yau three-folds. For phenomenological applications it is crucial to understand how these metrics vary across moduli space. In this work, we construct approximate analytic expressions for Ricci-flat Calabi-Yau metrics with explicit complex-structure and K\"ahler moduli dependence by combining machine-learned numerical data with symbolic regression. Our approach is based on an explicit Ansatz for the K\"ahler potential with moduli-dependent coefficients. Fitting this Ansatz to numerical data and applying symbolic regression allows us to reconstruct analytic formulae for these coefficients, thereby obtaining approximate Ricci-flat metrics with explicit moduli dependence. We apply the construction to a one-parameter family of bi-cubic three-folds in $\mathbb{P}^2 \times \mathbb{P}^2$, achieving percent-level agreement with the underlying numerical data.