Partial differential equation driven dynamic graph networks for predicting stream water temperature

This paper presents a physics-guided machine learning approach that incorporates partial differential equations (PDEs) in a graph neural network model to improve the prediction of water temperature in river networks. The standard graph neural network model often uses pre-defined edge weights based on distance or similarity measures. Such static graph structure can be limited in capturing multiple processes in a physical system that interact and evolve over time. The limitation to represent underlying physical processes can severely affect the performance of the predictive model, especially when we have access to limited training data. To better capture the dynamic interactions among multiple segments in a river network, we built a dynamic graph model, where the graph structure is driven by the PDE that describes underlying physical processes. We further combine the dynamic graph structure and the recurrent layers to model temporal dependencies and improve the prediction. We demonstrate the effectiveness of the proposed method in a subnetwork of the Delaware River Basin. In particular, we show that the proposed method outperforms existing physics-based and machine learning models in temperature prediction using sparse observation data for training. The proposed method has also been shown to produce better performance when generalized to different seasons.