Predicting vibrational displacement fields and mode shapes in thin plates is crucial for various engineering applications, particularly in the acoustic characterization of musical instrument soundboards. Traditional methods are limited by high computational costs or dense measurement requirements e.g., Finite Element Analysis (FEA). This study introduces a Physics-Informed Neural Network (PINN) approach to reconstruct displacement fields using sparse data. The PINN integrates the Kirchhoff plate equation into its training process, enabling accurate predictions even in data-sparse regions. The model was validated using COMSOL simulations of a thin rectangular plate, with material properties resembling a violin soundboard. Its performance was compared to Radial Basis Function (RBF) interpolation and data-driven neural networks. The PINN consistently outperformed the considered baselines, achieving robust results with minimal data, particularly for higher resonant frequencies where other methods fail.