<p>Reconstructing nonlinear waves from limited spatiotemporal measurement data and predicting their behavior in unsampled regions are crucial tasks in fields such as nonlinear optics, oceanography, and fluid dynamics. Experimental limitations and natural constraints often result in sparse data distributions across large areas, with denser measurements concentrated in specific regions. Furthermore, the lack of prior knowledge regarding the governing equations of wave motion introduces additional complexity. To address these challenges, we propose a hybrid deep learning framework that integrates a Forward-Backward Regression-based PDE-FIND (FBR-PDE-FIND) algorithm with a modified Physics-Informed Neural Network (PINN) architecture. This framework is designed to reconstruct and predict nonlinear waves, particularly solitons and rogue waves, from locally dense sparse datasets, without relying on prior knowledge of governing equations. We show how our approach can be applied to investigate various nonlinear waves, including bell-shaped solitons, kink-type solitons, bi-solitons, and rogue waves. The results validate the accuracy of the reconstructions, and also demonstrate the strong extrapolation capabilities of the trained neural networks, which enable accurate predictions of nonlinear waves beyond the sampled areas. Additionally, we observe that the governing equations for a specific wave may not be unique, allowing for the use of different equations to reconstruct and predict the same wave. This suggests the potential to investigate nonlinear waves using linear equations, offering an alternative perspective on wave analysis.</p>