This paper presents a study of physical Yukawa couplings in heterotic string compactifications on Calabi-Yau threefolds. The focus is on computing the normalized Yukawa couplings for the (27)^3 couplings, which describe the interactions of low-energy fields in the resulting four-dimensional N=1 theory. The calculation requires knowledge of the Ricci-flat metric, but in the standard embedding, where the vector bundle is the tangent bundle, the Weil–Petersson metric on the moduli space of complex structure deformations can be used instead. This metric is calculated without needing the Ricci-flat metric, allowing for the computation of normalized Yukawa couplings. The paper discusses three approaches to compute these couplings: (i) using the Kodaira–Spencer map, (ii) performing period integrals for cases with h^{1,1}=1, and (iii) using a machine-learned approximate Ricci-flat metric. The results from these methods agree, demonstrating the consistency of the approaches. The paper also presents numerical results for several Calabi-Yau manifolds, including the mirror of P^5[3,3], the quintic and Gepner model Y_{4;5}, and the Tian-Yau quotient. These results show good agreement with known exact results and highlight the effectiveness of the methods used. The paper also discusses the use of machine learning to approximate the Ricci-flat metric and to compute harmonic representatives of the vector bundle. These techniques allow for efficient computation of Yukawa couplings and are particularly useful for complex Calabi-Yau manifolds. The results demonstrate that the normalization of Yukawa couplings computed using the Weil–Petersson metric agree with those computed using the Ricci-flat metric, validating the methods used. The paper concludes that the methods presented provide a powerful tool for computing Yukawa couplings in heterotic string compactifications, with potential applications in string phenomenology. The results demonstrate the importance of understanding the complex geometry of Calabi-Yau manifolds and the role of the Weil–Petersson metric in computing physical couplings. The paper also highlights the potential of machine learning in improving the numerical methods used in string theory.This paper presents a study of physical Yukawa couplings in heterotic string compactifications on Calabi-Yau threefolds. The focus is on computing the normalized Yukawa couplings for the (27)^3 couplings, which describe the interactions of low-energy fields in the resulting four-dimensional N=1 theory. The calculation requires knowledge of the Ricci-flat metric, but in the standard embedding, where the vector bundle is the tangent bundle, the Weil–Petersson metric on the moduli space of complex structure deformations can be used instead. This metric is calculated without needing the Ricci-flat metric, allowing for the computation of normalized Yukawa couplings. The paper discusses three approaches to compute these couplings: (i) using the Kodaira–Spencer map, (ii) performing period integrals for cases with h^{1,1}=1, and (iii) using a machine-learned approximate Ricci-flat metric. The results from these methods agree, demonstrating the consistency of the approaches. The paper also presents numerical results for several Calabi-Yau manifolds, including the mirror of P^5[3,3], the quintic and Gepner model Y_{4;5}, and the Tian-Yau quotient. These results show good agreement with known exact results and highlight the effectiveness of the methods used. The paper also discusses the use of machine learning to approximate the Ricci-flat metric and to compute harmonic representatives of the vector bundle. These techniques allow for efficient computation of Yukawa couplings and are particularly useful for complex Calabi-Yau manifolds. The results demonstrate that the normalization of Yukawa couplings computed using the Weil–Petersson metric agree with those computed using the Ricci-flat metric, validating the methods used. The paper concludes that the methods presented provide a powerful tool for computing Yukawa couplings in heterotic string compactifications, with potential applications in string phenomenology. The results demonstrate the importance of understanding the complex geometry of Calabi-Yau manifolds and the role of the Weil–Petersson metric in computing physical couplings. The paper also highlights the potential of machine learning in improving the numerical methods used in string theory.