Deep learning is revolutionizing weather forecasting, with new data-driven models achieving accuracy on par with operational physical models for medium-term predictions. However, these models often lack interpretability, making their underlying dynamics difficult to understand and explain. This paper proposes methodologies to estimate the Koopman operator, providing a linear representation of complex nonlinear dynamics to enhance the transparency of data-driven models. Despite its potential, applying the Koopman operator to large-scale problems, such as atmospheric modeling, remains challenging. This study aims to identify the limitations of existing methods, refine these models to overcome various bottlenecks, and introduce novel convolutional neural network architectures that capture simplified dynamics.

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.