Learning more physically realistic dynamics in machine-learning based weather forecasting with latent-space constraints

Data-driven machine learning (ML) models are reshaping weather forecasting and have shown the potential to accelerate and surpass traditional physics-based approaches, leading to a second revolution in the field after data assimilation. However, most ML forecast models are trained with weighted variable-wise losses on rollout forecasts that neglect cross-variable and spatial error covariance induced by physical coupling, often yielding overly smooth and physically unrealistic long-range forecasts. To address this, we reformulate model training as a four-dimensional variational data assimilation (4DVar) problem that treats reanalysis data as imperfect observations. This enables the loss function to incorporate cross-variable error covariance structures that capture multivariate dependencies and their associated errors. In practice, we approximate this objective by computing the loss in an autoencoder-learned latent space of global atmospheric states. By encoding complex nonlinear couplings among atmospheric variables, this representation allows the high-dimensional, complex error covariance matrix in model space to be approximated as nearly diagonal in latent space, substantially simplifying implementation. We show that rollout training with latent-space constraints improves long-term forecast skill, while better preserving fine-scale structures and physical realism than the widely used model-space loss. Finally, we extend this framework to accommodate heterogeneous data sources, enabling the forecast model to be trained jointly on reanalysis and multi-source observations within a unified theoretical formulation.