stat.MLOct 23, 2026

Batched NN 4D-Var: a performant hybrid model framework for the Integrated Forecasting System

Authors: Alban Farchi, Marcin Chrust, Massimo Bonavita, Marc Bocquet, Wei Pan, Patrick Laloyaux

Organizations: ECMWF Bonn, Germany · ECMWF Reading, United Kingdom · CEREA, ENPC, EDF R&D Institut Polytechnique de Paris Île–de–France, France

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Abstract

Hybrid forecasting systems combining machine learning and numerical weather prediction offer a promising route toward improved forecast skill while preserving physical consistency. In this study, we investigate online learning of state-dependent model-error corrections within the Integrated Forecasting System using 4D-Var. Building on previous work, we represent model error using a neural network embedded within the forecast model and trained from observations through the data assimilation cycle. First, we introduce a targeted variant of neural-network 4D-Var in which only the neural-network parameters are estimated, while the atmospheric state is provided by a precomputed analysis. We then propose a batched extension that processes assimilation windows in random order and averages parameter increments over multiple windows, drawing inspiration from mini-batch stochastic optimization. Experiments are conducted with the Integrated Forecasting System of the European Centre for Medium-Range Weather Forecasts at an intermediate resolution. Sequential online learning substantially improves forecasts within the training period, with error reductions exceeding 10%in some cases. However, these improvements fail to generalize to unseen data. In contrast, the proposed batched formulation produces hybrid models that consistently improve forecast skill in out-of-sample experiments. Larger batch sizes accelerate convergence and improve robustness by reducing the noise of parameter updates. Diagnostics further show that the online-trained corrections preserve realistic forecast variability and do not improve skill through artificial smoothing. These results demonstrate that the formulation of the online learning problem is critical for generalisation and establish batched online training as a promising framework for hybrid machine-learning/data-assimilation systems.

Figures & tables

Figure 1: Illustration of the differences between 4D-Var (left), NN 4D-Var (middle), and targeted NN 4D-Var (right). In 4D-Var, the goal is to estimate the initial state x0\mathbf{x}_{0} only. NN 4D-Var provides a joint estimation of both initial state x0\mathbf{x}_{0} and model parameters p\mathbf{p} , while targeted NN 4D-Var focus on model parameters p\mathbf{p} only and relies on a pre-computed analysis x0pa\mathbf{x}^{\mathsf{pa}}_{0} for the initial state.
Figure 2: Illustration of an assimilation cycle in the batched targeted NN 4D-Var method. The cycle starts from a background for model parameters pb\mathbf{p}^{\mathsf{b}} . Random dates are sampled from the training dataset and assigned to a member in the batch. Each member then solves its own targeted NN 4D-Var problem, yielding individual parameter increments δpia\delta\mathbf{p}^{\mathsf{a}}_{i} , which are then averaged in a synchronisation step. An optional ML optimiser step follows (see section 4.3 ) before the parameter analysis pa\mathbf{p}^{\mathsf{a}} gets computed.
Figure 3: Normalised difference in RMSE between the forecasts of the offline-trained hybrid model and the control forecasts, as described in section 3.2.2 , for Z500 (top panels) and T850 (bottom panels) in the NH (right panels) and in the SH (left panels), evaluated over 2025.
Figure 4: Same as fig. 3 for the forecasts of the day described in section 3.3.2 .
Figure 5: Normalised difference in RMSE at day 55 between (i) the forecasts of the hybrid models trained using sequential online training, as described in section 3.3.3 , and (ii) the control forecasts as a function of online training time (number of assimilation cycles multiplied by the length of the assimilation window), for Z500 (top panels) and T850 (bottom panels) in the NH (right panels) and in the SH (left panels), evaluated over 2025. As the NN are first pre-trained offline, the scores at training time zero exactly correspond to the offline trained hybrid shown in the previous figures.
Figure 6: Same as fig. 5 at day 33 (in red) and at day 55 (in purple) for the hybrid models trained using non-sequential online training, as described in section 3.4.2 .
Figure 7: Total kinetic energy for vector winds at 500 hPa500\text{\,}\mathrm{hPa} for the control and hybrid forecasts at day 55 (left panel) and at day 1010 (right panel), averaged over the first two months of forecast (January to February 2025).
Figure 8: Summary scorecard comparing forecast RMSE differences between the hybrid model and the control forecasts in the 2025-48h setup, verified against operational analysis and against independent observations. Each box corresponds to a 24 h24\text{\,}\mathrm{h} lead-time interval from forecast day 11 to day 1010 . The colour saturation is proportional to the magnitude of the normalised differences, with blue meaning improvement relative to control and red the other way around.
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