Optimizing Yukawa couplings to suppress Dimension-five Proton Decay in $SU(5)$ GUT

The minimal supersymmetric $SU(5)$ grand unified theory (GUT) provides a highly compelling framework for physics beyond the Standard Model (SM). However, it suffers from a severe phenomenological challenge: rapid proton decay mediated by colored-Higgsino exchange via dimension-five operators. Resolving this issue often requires adjustments to the Yukawa couplings and the potential sectors, generating a vast and complex parameter space where traditional brute-force numerical scans are rendered computationally intractable due to the curse of dimensionality. In this paper, we overcome this limitation by applying machine learning optimization techniques. We investigate a supersymmetric $SU(5)$ model extended with $\mathbf{45}$ and $\overline{\mathbf{45}}$ Higgs representations, defining a loss function based on the partial decay width of $p \to K^+ \bar{\nu}$. Utilizing the Adam optimizer, we systematically explore the 33-dimensional parameter space to identify regions that suppress proton decay. Furthermore, we vary $\tan \beta$ to thoroughly investigate whether the optimized proton lifetime can consistently exceed the stringent experimental lower bound of $5.9 \times 10^{33}$ years established by the Super-Kamiokande collaboration.