# Using machine learning to correct model error in data assimilation and forecast applications

arXiv: https://arxiv.org/abs/2010.12605 · HTML: https://mlsanity.com/paper/2010.12605/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2020.

## Built on by (most cited first, 30 of 41)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [A posteriori learning for quasi-geostrophic turbulence parametrization](https://mlsanity.com/paper/2204.03911/) (2022)
- [Variational Data Assimilation with a Learned Inverse Observation Operator](https://mlsanity.com/paper/2102.11192/) (2021)
- [Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models](https://mlsanity.com/paper/2204.03497/) (2022)
- [FengWu-4DVar: Coupling the Data-driven Weather Forecasting Model with 4D Variational Assimilation](https://mlsanity.com/paper/2312.12455/) (2023)
- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [Reduced-Order Autodifferentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2301.11961/) (2023)
- [A comparison of combined data assimilation and machine learning methods for offline and online model error correction](https://mlsanity.com/paper/2107.11114/) (2021)
- [Neural Koopman prior for data assimilation](https://mlsanity.com/paper/2309.05317/) (2023)
- [Latent assimilation with implicit neural representations for unknown dynamics](https://mlsanity.com/paper/2309.09574/) (2023)
- [State, global and local parameter estimation using local ensemble Kalman filters: applications to online machine learning of chaotic dynamics](https://mlsanity.com/paper/2107.11253/) (2021)
- [Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case](https://mlsanity.com/paper/2110.00546/) (2021)
- [Gradient-free online learning of subgrid-scale dynamics with neural emulators](https://mlsanity.com/paper/2310.19385/) (2023)
- [Data Assimilation in Chaotic Systems Using Deep Reinforcement Learning](https://mlsanity.com/paper/2401.00916/) (2024)
- [Ensemble Kalman filter in latent space using a variational autoencoder pair](https://mlsanity.com/paper/2502.12987/) (2025)
- [Align-DA: Align Score-based Atmospheric Data Assimilation with Multiple Preferences](https://mlsanity.com/paper/2505.22008/) (2025)
- [Combining data assimilation and machine learning to estimate parameters of a convective-scale model](https://mlsanity.com/paper/2109.02953/) (2021)
- [Machine-learning parameter tracking with partial state observation](https://mlsanity.com/paper/2311.09142/) (2023)
- [Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilation](https://mlsanity.com/paper/2402.15656/) (2024)
- [Foundation Models for Environmental Science: A Survey of Emerging Frontiers](https://mlsanity.com/paper/2504.04280/) (2025)
- [Learning 4DVAR inversion directly from observations](https://mlsanity.com/paper/2211.09741/) (2022)
- [Combined Optimization of Dynamics and Assimilation with End-to-End Learning on Sparse Observations](https://mlsanity.com/paper/2409.07137/) (2024)
- [Online machine-learning forecast uncertainty estimation for sequential data assimilation](https://mlsanity.com/paper/2305.08874/) (2023)
- [Development of an offline and online hybrid model for the Integrated Forecasting System](https://mlsanity.com/paper/2403.03702/) (2024)
- [Decomposing weather forecasting into advection and convection with neural networks](https://mlsanity.com/paper/2405.06590/) (2024)
- [Ensemble Transport Filter via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/2407.11518/) (2024)
- [RL-DAUNCE: Reinforcement Learning-Driven Data Assimilation with Uncertainty-Aware Constrained Ensembles](https://mlsanity.com/paper/2505.05452/) (2025)
- [Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems](https://mlsanity.com/paper/2412.14318/) (2024)

## Abstract (arXiv)

The idea of using machine learning (ML) methods to reconstruct the dynamics of a system is the topic of recent studies in the geosciences, in which the key output is a surrogate model meant to emulate the dynamical model. In order to treat sparse and noisy observations in a rigorous way, ML can be combined to data assimilation (DA). This yields a class of iterative methods in which, at each iteration a DA step assimilates the observations, and alternates with a ML step to learn the underlying dynamics of the DA analysis. In this article, we propose to use this method to correct the error of an existent, knowledge-based model. In practice, the resulting surrogate model is an hybrid model between the original (knowledge-based) model and the ML model. We demonstrate numerically the feasibility of the method using a two-layer, two-dimensional quasi-geostrophic channel model. Model error is introduced by the means of perturbed parameters. The DA step is performed using the strong-constraint 4D-Var algorithm, while the ML step is performed using deep learning tools. The ML models are able to learn a substantial part of the model error and the resulting hybrid surrogate models produce better short- to mid-range forecasts. Furthermore, using the hybrid surrogate models for DA yields a significantly better analysis than using the original model.