FERA: Forecast-Conditioned Error Representations for Data Assimilation

Data assimilation is fundamental to weather prediction because forecasts must be continually corrected using observations that constrain only a small fraction of the atmospheric state. The central challenge is not merely fitting these observations, but propagating their information to unobserved locations and variables through a plausible representation of forecast uncertainty. Classical climatological covariances are stable but cannot adapt their correction geometry to the current flow, while small ensembles provide flow dependence only through a rank-limited sample. Neural methods offer more expressive representations, but often perform assimilation directly in learned latent spaces or retain nonlinear neural decoders inside the online optimization, making the connection between observations, physical corrections, and inference less explicit. We introduce FERA (Forecast-conditioned Error Representation for Assimilation), which separates learned error representation from online Bayesian inference. A forecast-conditioned neural transform constructs multivariate physical-space correction patterns, while held-out-variable and sparse-observation objectives train these patterns to recover errors from partial information. During assimilation, the neural transform is frozen: observations determine the correction coefficients through a linear-Gaussian Bayesian update, while the ensemble refines their prior uncertainty and a full-rank historical component preserves directions beyond the member span. For linear observations, this yields a unique, initialization-free analysis with an explicit observation-to-correction map, while retaining forecast-adaptive error structure across cycles. In month-long global cycling experiments with a 69-channel AI weather model, FERA outperforms the strongest matched baseline in 46 of 48 ensemble-size-diagnostic comparisons. In a fixed-input replay, FERA shows lower numerical sensitivity to equivalent observation reorderings than the tested nonlinear neural variational baseline, without optimizer-initialization dependence. Ablation studies further demonstrate the effectiveness of both forecast conditioning and partial-information training for sustained cycling. Together, these results link forecast-adaptive error learning to an explicit Bayesian analysis, providing a basis for reliable and interpretable assimilation.