Systematic Study on the $\alpha$-particle preformation factor in the theory of $\alpha$-decay based on the Tabular Prior-data Fitted Network (TabPFN)

A hybrid approach combining the Tabular Prior-data Fitted Network (TabPFN) with the Coulomb and Proximity Potential Model (CPPM) is developed to investigate $\alpha$-particle preformation factors $P_{\alpha}$ and their impact on $\alpha$-decay half-lives. The TabPFN model, trained on 498 nuclei, accurately learns the relationship between nuclear structure properties and $P_{\alpha}$, achieving a root mean square deviation of $\sigma_{\mathrm{rms}} = 0.211$. The predicted factors reveal clear odd-even staggering and shell closure effects, and exhibit linear correlations with both $Q_{\alpha}^{-1/2}$ and the fragmentation potential $V_{\mathrm{frag}}$. When incorporated into CPPM calculations, the machine-learning-based $P_{\alpha}$ values significantly improve half-life predictions. Similar improvements are also obtained when deformation effects are included in the potential barrier description. The capability of the model is further demonstrated through predictions for superheavy nuclei ($Z = 117$--120), suggesting $N = 184$ as a potential neutron magic number.