Synchronous generators (SGs) are the cornerstone of modern power systems. However, achieving accurate dynamic modeling of SGs, particularly when considering their complex nonlinear characteristics, has been a persistent challenge for over a century. Neural networks are a promising alternative for SG dynamic modeling, but their training typically lacks sufficient data. To address this, a neural network-based approach for SG dynamic modeling using data augmentation is proposed. The proposed method employs an improved recurrent neural network (RNN) and a practical two-stage learning strategy. In data augmentation and initial training stage, comprehensive data augmentation is performed using physics-based simulations for initial training, and the tailored improved RNN architecture further enables the model to effectively learn and capture dynamics that closely align with physical principles. In measurement-driven fine-tuning stage, scarce real-world measurement data from an in-service generator are used to fine-tune the model, further enhancing its accuracy in real-world operating conditions. Following initial training, the proposed model exhibits generalization ability across diverse fault scenarios, including challenging worst-case and marginal stabilization conditions, accurately replicating physical principles to ensure baseline accuracy and further validating its reliability. Finally, the proposed model achieves a significant relative error reduction compared to the state-of-the-art SG dynamic model, GENQEC, highlighting its potential as a superior alternative for precise SG dynamic representation.