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The solar dynamo relies on the regeneration of the poloidal magnetic field through processes strongly modulated by nonlinear feedbacks such as tilt quenching (TQ) and latitude quenching (LQ). These mechanisms play a decisive role in regulating the buildup of the Sun's polar field and, in turn, the amplitude of future solar cycles. In this work, we employ Physics-Informed Neural Networks (PINN) to solve the surface flux transport (SFT) equation, embedding physical constraints directly into the neural network framework. By systematically varying transport parameters, we isolate the relative contributions of TQ and LQ to polar dipole buildup. We use the residual dipole moment as a diagnostic for cycle-to-cycle amplification and show that TQ suppression strengthens with increasing diffusivity, while LQ dominates in advection-dominated regimes. The ratio $\Delta D_{\mathrm{LQ}}/\Delta D_{\mathrm{TQ}}$ exhibits a smooth inverse-square dependence on the dynamo effectivity range, refining previous empirical fits with improved accuracy and reduced scatter. The results further reveal that the need for a decay term is not essential for PINN set-up due to the training process. Compared with the traditional 1D SFT model, the PINN framework achieves significantly lower error metrics and more robust recovery of nonlinear trends. Our results suggest that the nonlinear interplay between LQ and TQ can naturally produce alternations between weak and strong cycles, providing a physical explanation for the observed even-odd cycle modulation. These findings demonstrate the potential of PINN as an accurate, efficient, and physically consistent tool for solar cycle prediction.