# A comparison of combined data assimilation and machine learning methods for offline and online model error correction

arXiv: https://arxiv.org/abs/2107.11114 · HTML: https://mlsanity.com/paper/2107.11114/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 45 references that are in the map.
- First posted on arXiv 23 Jul 2021.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [ReZero is All You Need: Fast Convergence at Large Depth](https://mlsanity.com/paper/2003.04887/) (2020)
- [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Physics Guided RNNs for Modeling Dynamical Systems: A Case Study in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/1810.13075/) (2018)
- [Machine Learning for Stochastic Parameterization: Generative Adversarial Networks in the Lorenz '96 Model](https://mlsanity.com/paper/1909.04711/) (2019)
- [Using machine learning to correct model error in data assimilation and forecast applications](https://mlsanity.com/paper/2010.12605/) (2020)
- [Combining data assimilation and machine learning to infer unresolved scale parametrisation](https://mlsanity.com/paper/2009.04318/) (2020)
- [Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization](https://mlsanity.com/paper/2001.06270/) (2020)
- [Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation](https://mlsanity.com/paper/2007.07383/) (2020)
- [Combining Machine Learning with Knowledge-Based Modeling for Scalable Forecasting and Subgrid-Scale Closure of Large, Complex, Spatiotemporal Systems](https://mlsanity.com/paper/2002.05514/) (2020)
- [Applying machine learning to improve simulations of a chaotic dynamical system using empirical error correction](https://mlsanity.com/paper/1904.10904/) (2019)
- [Online learning of both state and dynamics using ensemble Kalman filters](https://mlsanity.com/paper/2006.03859/) (2020)
- [Machine Learning, Deepest Learning: Statistical Data Assimilation Problems](https://mlsanity.com/paper/1707.01415/) (2017)

## Built on by (most cited first, 13 of 13)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Integrating Recurrent Neural Networks with Data Assimilation for Scalable Data-Driven State Estimation](https://mlsanity.com/paper/2109.12269/) (2021)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [Efficient deep data assimilation with sparse observations and time-varying sensors](https://mlsanity.com/paper/2310.16187/) (2023)
- [Evaluation of Machine Learning Techniques for Forecast Uncertainty Quantification](https://mlsanity.com/paper/2111.14844/) (2021)
- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilation](https://mlsanity.com/paper/2404.06749/) (2024)
- [Adjoint-Matching Neural Network Surrogates for Fast 4D-Var Data Assimilation](https://mlsanity.com/paper/2111.08626/) (2021)
- [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)
- [Integrating Score-Based Diffusion Models with Machine Learning-Enhanced Localization for Advanced Data Assimilation in Geological Carbon Storage](https://mlsanity.com/paper/2511.05266/) (2025)
- [The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling](https://mlsanity.com/paper/2605.31547/) (2026)
- [Composing a surrogate observation operator for sequential data assimilation](https://mlsanity.com/paper/2201.12514/) (2022)

## Abstract (arXiv)

Recent studies have shown that it is possible to combine machine learning methods with data assimilation to reconstruct a dynamical system using only sparse and noisy observations of that system. The same approach can be used to correct the error of a knowledge-based model. The resulting surrogate model is hybrid, with a statistical part supplementing a physical part. In practice, the correction can be added as an integrated term (i.e. in the model resolvent) or directly inside the tendencies of the physical model. The resolvent correction is easy to implement. The tendency correction is more technical, in particular it requires the adjoint of the physical model, but also more flexible. We use the two-scale Lorenz model to compare the two methods. The accuracy in long-range forecast experiments is somewhat similar between the surrogate models using the resolvent correction and the tendency correction. By contrast, the surrogate models using the tendency correction significantly outperform the surrogate models using the resolvent correction in data assimilation experiments. Finally, we show that the tendency correction opens the possibility to make online model error correction, i.e. improving the model progressively as new observations become available. The resulting algorithm can be seen as a new formulation of weak-constraint 4D-Var. We compare online and offline learning using the same framework with the two-scale Lorenz system, and show that with online learning, it is possible to extract all the information from sparse and noisy observations.