For computational fluid dynamics, there is a considerable interest in using neural networks for accelerating simulations. However, these learning-based models suffer from scalability issues when training on high-dimensional and high-resolution simulation data generated for real-world applications. In this work, we study the problem of improving accuracy of desired physical properties using graph learning models for learning complex fluid dynamics, while operating on mesh-reduced space. We design several tailored modules to incorporate physical-informed knowledge into a two-stage prediction model, which directs the learning process to focus more on the region of interest (ROI). Prediction will then be made in a mesh-reduced space, which helps reduce computational costs while preserving important physical properties. Results on simulated unsteady fluid flow data show that even under reduced operational space, our method still achieves desirable performance on accuracy and generalizability of both prediction and physical consistency over region of interests.