Sparsely available data points cause numerical error on finite differences which hinders us from modeling the dynamics of physical systems. The discretization error becomes even larger when the sparse data are irregularly distributed or defined on an unstructured grid, making it hard to build deep learning models to handle physics-governing observations on the unstructured grid. In this paper, we propose a novel architecture, Physics-aware Difference Graph Networks (PA-DGN), which exploits neighboring information to learn finite differences inspired by physics equations. PA-DGN leverages data-driven end-to-end learning to discover underlying dynamical relations between the spatial and temporal differences in given sequential observations. We demonstrate the superiority of PA-DGN in the approximation of directional derivatives and the prediction of graph signals on the synthetic data and the real-world climate observations from weather stations.