Semiclassical mapping dynamics offer a computationally tractable approach for simulating nonadiabatic processes in complex molecular systems. This work presents a comparative study of purely data-driven (DD) neural networks and physics-informed neural networks (PINNs) for learning the Markovian propagation of trajectories within the Meyer-Miller-Stock-Thoss (MMST) mapping Hamiltonian framework. Using the spin-boson model as a benchmark, we assess the performance of both approaches in reproducing the nonadiabatic dynamical details for classical mapping model (CMM) and symmetrical quasiclassical (SQC) dynamics. Our results demonstrate that PINNs, which explicitly incorporate the physical equations of motion, significantly outperform DD models, especially when trained with limited datasets. PINNs accurately capture the population dynamics and preserve the physical correlations in phase space, regardless of the underlying neural network architecture (fully connected network or gated recurrent unit) or the specific MMST Hamiltonian formulation. In contrast, DD models exhibit substantial inaccuracies and unphysical behaviors. This clearly shows the key benefit of embedding physical laws into machine learning frameworks to achieve data-efficient, accurate, and reliable simulations of nonadiabatic phenomena.