Data-Driven Initial Guess Selection for Numerical Weather Prediction Solvers

Recent advancements in weather forecasting have been driven by modern AI models like GraphCast, which significantly enhance predictive accuracy. However, the interpretability of these models remains a challenge, as they often function as opaque "black boxes" that obscure the reasoning behind their predictions. In contrast, traditional Numerical Weather Prediction (NWP) models, which are grounded in physical laws and offer complete transparency, face limitations in computational efficiency due to their high dimensionality and the complexity of solving large linear systems. We explore two distinct dynamical systems—shallow-water and Gaussian-bubble—and investigate methods to improve the convergence of iterative solvers used in forecasting these systems. Specifically, we propose leveraging Dynamic Mode Decomposition (DMD) to generate more accurate initial guesses for the GMRES solver, alongside two DMD-inspired approaches: Fixed Previous and Random Previous. Additionally, we introduce a probabilistic linear combination approach. Our results demonstrate faster convergence rates in iterative solvers, albeit with the emergence of biases in lower tolerance solutions.