Leveraging transport theory to advance machine learning for weather forecasting

Transport processes govern the evolution of many physical systems, including atmospheric dynamics, where information, mass, energy, and uncertainty propagate across space and time. Although machine learning has achieved remarkable success in forecasting complex spatiotemporal phenomena, many existing models rely on local operators and pointwise objectives that inadequately capture the transport mechanisms underlying physical processes. This limitation is particularly pronounced in weather forecasting, where accurate prediction requires modeling both the movement of meteorological structures and the evolution of uncertainty across multiple spatial and temporal scales. This thesis investigates how principles from transport theory can be leveraged to develop machine learning models that are more accurate and physically grounded. The central premise of this thesis is that transport provides a unifying framework for learning spatiotemporal dynamics. Rather than treating forecasting as a purely statistical prediction problem, we develop learning architectures that explicitly model the transport of information, features, and probability distributions. Building on concepts from partial differential equations, dynamical systems, and generative modeling, we propose transport-aware methods for both deterministic and probabilistic forecasting. The contributions of this thesis are twofold. First, we introduce transport-aware neural architectures based on advection-diffusion-reaction dynamics, incorporating a semi-Lagrangian transport operator that enables efficient non-local information propagation in convolutional neural networks. Second, the state-of-the-art probabilistic models are based on stochastic diffusion models that are expensive and also not naturally suitable for sensitive dynamical systems. We extend transport concepts to probabilistic forecasting by formulating uncertainty quantification as a distributional transport problem and developing flow-matching-based methods that learn transport maps between probability distributions, enabling faster generation of physically consistent ensemble forecasts. Experiments on benchmark spatiotemporal datasets, including weather forecasting applications, demonstrate improvements in predictive accuracy, long-range information propagation, uncertainty calibration, and physical consistency. Collectively, the results establish transport theory as a powerful organizing principle for machine learning and provide a general framework for incorporating physically motivated transport mechanisms into forecasting systems and other scientific machine learning applications.