用符号回归找到刻画卡拉比-丘度量的简洁公式。
Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation
- 以平衡度量为教师，用符号回归拟合度量偏差
- 仅用两个对称特征即达94.6%解释力，三阶多项式达99.94%精度
- 公式跨模空间稳定可用，适合几何与物理交叉研究者
点态行列式比值 $ R_ψ(z) = \log\left(\frac{\det g_{\mathrm{RF}}(z;ψ)}{\det g_{\mathrm{FS}}(z)}\right) $ 衡量了Dwork五次超曲面的里奇平坦度量偏离法博尼-施图迪基准的程度。我们探究该标量可观测量是否可用少量射影不变量紧凑描述，且其结构在复结构模空间中是否通用。以Donaldson的 $k=10$ 平衡度量作为代数教师，结合采样点上的符号回归，发现在此受限的仅含模参数的特征空间中，两个低阶对称特征——幂和 $p_2 = \sum_i |z_i|^4$ 与三次初等对称多项式 $σ_3 = e_3$——已能捕捉大部分教师变化。一个关于 $(p_2, σ_3)$ 的三阶多项式在留出测试集上达到 $R^2=0.946$；加入其余低阶对称生成元后，提升不足 $10^{-3}$。在同一双特征空间中，符号回归识别出一个五项有理多项式表达式，与 $k=10$ 教师匹配度高达 $R^2=0.9994$。在 $ψ∈[0,0.8]$ 范围内重拟合同一函数框架，点云上平均行列式比值 $\langle R_ψ\rangle$ 与局部教师偏差小于 $0.01\%$，且拟合系数平滑变化。全纯尤卡瓦耦合 $κ_{111}=5$ 作为归一化检验被重现。整体结果提供了一种对Dwork族中度量导出标量可观测量的紧凑符号描述，但受制于有限 $k$ 的教师度量，而非推导出闭合形式的里奇平坦度量。
原文摘要 · Abstract (English)
The pointwise determinant ratio \[ R_ψ(z)\equiv \log\!\left(\frac{\det g_{\mathrm{RF}}(z;ψ)}{\det g_{\mathrm{FS}}(z)}\right) \] measures how the Ricci-flat metric on the Dwork quintic departs from the Fubini--Study baseline. We ask whether this scalar observable can be described compactly in terms of a small number of projective invariants, and whether the same scaffold remains usable across complex-structure moduli. Using Donaldson's $k=10$ balanced metric as an algebraic teacher and symbolic regression on sampled points, we find that, within the restricted moduli-only feature class studied here, two low-order symmetric features, the power sum $p_2=\sum_i |z_i|^4$ and the cubic elementary symmetric polynomial $σ_3=e_3$, already capture most of the teacher variation. A degree-3 polynomial in $(p_2,σ_3)$ achieves held-out test $R^2=0.946$, while adding the remaining low-order symmetric generators changes this by less than $10^{-3}$. Within the same two-feature space, symbolic regression identifies a five-term rational-polynomial expression that matches the $k=10$ teacher with $R^2=0.9994$. Refitting the same functional scaffold across $ψ\in[0,0.8]$ keeps the mean determinant-ratio proxy $\langle R_ψ\rangle$ within $0.01\%$ of the local teachers on the sampled point clouds and yields smoothly varying fitted coefficients over the studied range. The holomorphic Yukawa coupling $κ_{111}=5$ is reproduced as a normalization check only. Taken together, these results provide a compact symbolic description of one metric-derived scalar observable on the Dwork family, while remaining bounded by the finite-$k$ teacher used for distillation rather than establishing a closed-form Ricci-flat metric.
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