mlsanityData to 7 Oct 2026

Online model error correction with neural networks in the incremental 4D-Var framework

arXiv 2210.13817 ↗First posted 25 Oct 2022stat.ML cs.LGcited by 13

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Abstract (arXiv)

Recent studies have demonstrated that it is possible to combine machine learning with data assimilation to reconstruct the dynamics of a physical model partially and imperfectly observed. Data assimilation is used to estimate the system state from the observations, while machine learning computes a surrogate model of the dynamical system based on those estimated states. The surrogate model can be defined as an hybrid combination where a physical model based on prior knowledge is enhanced with a statistical model estimated by a neural network. The training of the neural network is typically done offline, once a large enough dataset of model state estimates is available. By contrast, with online approaches the surrogate model is improved each time a new system state estimate is computed. Online approaches naturally fit the sequential framework encountered in geosciences where new observations become available with time. In a recent methodology paper, we have developed a new weak-constraint 4D-Var formulation which can be used to train a neural network for online model error correction. In the present article, we develop a simplified version of that method, in the incremental 4D-Var framework adopted by most operational weather centres. The simplified method is implemented in the ECMWF Object-Oriented Prediction System, with the help of a newly developed Fortran neural network library, and tested with a two-layer two-dimensional quasi geostrophic model. The results confirm that online learning is effective and yields a more accurate model error correction than offline learning. Finally, the simplified method is compatible with future applications to state-of-the-art models such as the ECMWF Integrated Forecasting System.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 21 of its 64 references that are in the map. First posted on arXiv 25 Oct 2022.

Builds on

21 of 64 references.

References · 21

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9 more
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 4; 2024: 6; 2025: 2; 2026: 1.

13 papers.

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  1. 2023Machine learning with data assimilation and uncertainty quantification for dynamical systems: a reviewcited by 74
  2. 2024Deep Generative Data Assimilation in Multimodal Settingcited by 28
  3. 2023AI Foundation Models for Weather and Climate: Applications, Design, and Implementationcited by 19
  4. 2024CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilationcited by 7
  5. 2024CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilationcited by 6
  6. 2024Joint Parameter and Parameterization Inference with Uncertainty Quantification through Differentiable Programmingcited by 5
  7. 2024TorchDA: A Python package for performing data assimilation with deep learning forward and transformation functionscited by 5
  8. 2025Ensemble Kalman filter in latent space using a variational autoencoder paircited by 5
  9. 2023Online Calibration of Deep Learning Sub-Models for Hybrid Numerical Modeling Systemscited by 4
  10. 2023Online machine-learning forecast uncertainty estimation for sequential data assimilationcited by 2
  11. 2024Development of an offline and online hybrid model for the Integrated Forecasting Systemcited by 1
  12. 2025PnP-DA: Towards Principled Plug-and-Play Integration of Variational Data Assimilation and Generative Modelscited by 0
1 more
  1. 2026Sparsely-Supervised Data Assimilation via Physics-Informed Schr"odinger Bridgecited by 0