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.

Issues of fairness often arise in graphical neural networks used for misinformation detection. However, improving fairness can often come at the cost of reducing accuracy and vice versa. Therefore, we formulate the task of balancing accuracy and fairness as a multi-objective optimization (MOO) problem where we seek to find a set of Pareto optimal solutions. Traditional first-order approaches to solving MOO problems such as multi-gradient descent can be costly, especially with large neural networks. Instead, we describe a more efficient approach using the predictor-corrector method. Given an initial Pareto optimal point, this approach predicts the direction of a neighboring solution and refines this prediction using a few steps of multi-gradient descent. We show experimentally that this approach allows for the generation of high-quality Pareto fronts faster than baseline optimization methods.

Multiple-objective optimization (MOO) aims to simultaneously optimize multiple conflicting o bjectives a nd has found important applications in machine learning, such as simultaneously minimizing classification a nd f airness l osses. At an optimum, further optimizing one objective will necessarily increase at least another objective, and decision-makers need to comprehensively explore multiple optima to pin-point one final solution. We address the efficiency of exploring the Pareto front that contains all optima. First, stochastic multi-gradient descent (SMGD) takes time to converge to the Pareto front with large neural networks and datasets. Instead, we explore the Pareto front as a manifold from a few initial optima, based on a predictor-corrector method. Second, for each exploration step, the predictor iteratively solves a large-scale linear system that scales quadratically in the number of model parameters, and requires one backpropagation to evaluate a second-order Hessian-vector product per iteration of the solver. We propose a Gauss-Newton approximation that scales linearly, and that requires only first-order i nner-product p er i teration. T hird, we explore different linear system solvers, including the MINRES and conjugate gradient methods for approximately solving the linear systems. The innovations make predictor-corrector efficient for large networks and datasets. Experiments on a fair misinformation detection task show that 1) the predictor-corrector method can find Pareto fronts better than or similar to SMGD with less time, and 2) the proposed first-order method does not harm the quality of the Pareto front identified b y t he second-order method, while further reducing running time.