A novel physics-aware graph network using high-order numerical methods in weather forecasting model

Abstract

In this paper, we present a novel architecture for accurate weather forecasting within the framework of Physics-aware Graph Networks (PaGN), which aims to improve the prediction of weather time series by integrating both data and physical equations defined on sparsely distributed spatial domains. This approach employs geometric deep learning by accounting for both supervised loss and physics loss. Solely in supervised learning, data-driven training demands a large amount of data, whereas integrating physics information can reduce this requirement, resulting in faster training. The physics awareness also provides a stronger inductive bias and better generalization, directing the machine learning process by clarifying the complexity of weather data. Thus, a more accurate discretization of physics equations can enhance the physics awareness of the complex weather dynamics. Meanwhile, weather data is essentially gathered at observatories that are not located densely, since covering the Earth with numerous observatories is prohibitively expensive. Given its reliance on graph structure, the proposed PaGN is well-suited for handling sparsely distributed data particularly defined on arbitrary geometric domain. One of the main contributions of this study is the use of high-order numerical methods approximating physics equations defined on graph neural networks for achieving accuracy improvements. The fractional graph Laplacian operator is further integrated with the physics equations to account for the non-local characteristics of complex weather data. For computational efficiency, the accuracy improved PaGN architecture is smoothly implemented in the embedding space. A series of numerical tests on weather datasets is performed to verify the improved accuracy, robustness, and applicability of the proposed methodology. We first carry out the numerical study on the synthetic datasets obtained from the physics equations with an extra forcing term. We then extensively investigate the accuracies of weather forecasting for various configurations in terms of the geophysical regions and the accuracy-improved PaGN approaches for diffusion and wave equations. Moreover, we emphasize that the fractional graph Laplacian operator incorporated into our PaGN model can further improve the accuracy for weather forecasting.