The paper discusses the computation of physical Yukawa couplings in heterotic string compactifications on Calabi–Yau threefolds. The authors focus on the "standard embedding" of the heterotic $E_8 \times E_8$ superstring theory, where the vector bundle $V$ is taken to be the tangent bundle of the Calabi–Yau manifold. They introduce the Weil–Petersson metric on the moduli space of complex structure deformations of the Calabi–Yau manifold, which allows them to compute normalized Yukawa couplings without needing the Ricci-flat metric. This metric can be calculated using the Kodaira–Spencer map and period integrals for Calabi–Yau spaces with $h^{2,1} = 1$. The authors compare these methods with machine learning-based approximations of the Ricci-flat metric and find excellent agreement. They apply these methods to various Calabi–Yau manifolds, including the Fermat quintic, the intersection of two cubics in $\mathbb{P}^5$, and the Tian–Yau manifold, and demonstrate the correctness of their normalization techniques. The paper also includes a discussion on the numerical implementation of the Weil–Petersson metric and the use of machine learning for harmonic form computation.The paper discusses the computation of physical Yukawa couplings in heterotic string compactifications on Calabi–Yau threefolds. The authors focus on the "standard embedding" of the heterotic $E_8 \times E_8$ superstring theory, where the vector bundle $V$ is taken to be the tangent bundle of the Calabi–Yau manifold. They introduce the Weil–Petersson metric on the moduli space of complex structure deformations of the Calabi–Yau manifold, which allows them to compute normalized Yukawa couplings without needing the Ricci-flat metric. This metric can be calculated using the Kodaira–Spencer map and period integrals for Calabi–Yau spaces with $h^{2,1} = 1$. The authors compare these methods with machine learning-based approximations of the Ricci-flat metric and find excellent agreement. They apply these methods to various Calabi–Yau manifolds, including the Fermat quintic, the intersection of two cubics in $\mathbb{P}^5$, and the Tian–Yau manifold, and demonstrate the correctness of their normalization techniques. The paper also includes a discussion on the numerical implementation of the Weil–Petersson metric and the use of machine learning for harmonic form computation.