# Online machine-learning forecast uncertainty estimation for sequential data assimilation

arXiv: https://arxiv.org/abs/2305.08874 · HTML: https://mlsanity.com/paper/2305.08874/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 46 references that are in the map.
- First posted on arXiv 12 May 2023.

## Builds on (most cited first, 11 of 11)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [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)
- [Deep Learning for Post-Processing Ensemble Weather Forecasts](https://mlsanity.com/paper/2005.08748/) (2020)
- [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)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [A comparison of combined data assimilation and machine learning methods for offline and online model error correction](https://mlsanity.com/paper/2107.11114/) (2021)
- [Analog ensemble data assimilation and a method for constructing analogs with variational autoencoders](https://mlsanity.com/paper/2006.01101/) (2020)
- [Evaluation of Machine Learning Techniques for Forecast Uncertainty Quantification](https://mlsanity.com/paper/2111.14844/) (2021)
- [Predicting Weather Uncertainty with Deep Convnets](https://mlsanity.com/paper/1911.00630/) (2019)
- [Accuracy-Reliability Cost Function for Empirical Variance Estimation](https://mlsanity.com/paper/1803.04475/) (2018)

## Built on by (most cited first, 2 of 2)

- [Balancing Accuracy and Speed: A Multi-Fidelity Ensemble Kalman Filter with a Machine Learning Surrogate Model](https://mlsanity.com/paper/2512.12276/) (2025)
- [Hybrid machine learning data assimilation for marine biogeochemistry](https://mlsanity.com/paper/2504.05218/) (2025)

## Abstract (arXiv)

Quantifying forecast uncertainty is a key aspect of state-of-the-art numerical weather prediction and data assimilation systems. Ensemble-based data assimilation systems incorporate state-dependent uncertainty quantification based on multiple model integrations. However, this approach is demanding in terms of computations and development. In this work a machine learning method is presented based on convolutional neural networks that estimates the state-dependent forecast uncertainty represented by the forecast error covariance matrix using a single dynamical model integration. This is achieved by the use of a loss function that takes into account the fact that the forecast errors are heterodastic. The performance of this approach is examined within a hybrid data assimilation method that combines a Kalman-like analysis update and the machine learning based estimation of a state-dependent forecast error covariance matrix. Observing system simulation experiments are conducted using the Lorenz'96 model as a proof-of-concept. The promising results show that the machine learning method is able to predict precise values of the forecast covariance matrix in relatively high-dimensional states. Moreover, the hybrid data assimilation method shows similar performance to the ensemble Kalman filter outperforming it when the ensembles are relatively small.