# State, global and local parameter estimation using local ensemble Kalman filters: applications to online machine learning of chaotic dynamics

arXiv: https://arxiv.org/abs/2107.11253 · HTML: https://mlsanity.com/paper/2107.11253/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 44 references that are in the map.
- First posted on arXiv 23 Jul 2021.

## Builds on (most cited first, 9 of 9)

- [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)
- [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)
- [Online learning of both state and dynamics using ensemble Kalman filters](https://mlsanity.com/paper/2006.03859/) (2020)
- [Reservoir Computing as a Tool for Climate Predictability Studies](https://mlsanity.com/paper/2103.06206/) (2021)
- [Bilinear residual Neural Network for the identification and forecasting of dynamical systems](https://mlsanity.com/paper/1712.07003/) (2017)

## Built on by (most cited first, 7 of 7)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [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)
- [Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network](https://mlsanity.com/paper/2507.08749/) (2025)
- [Development of an offline and online hybrid model for the Integrated Forecasting System](https://mlsanity.com/paper/2403.03702/) (2024)
- [Accounting for model error in ensemble variational data assimilation through learned generative dynamics](https://mlsanity.com/paper/2610.17333/) (2026)

## Abstract (arXiv)

In a recent methodological paper, we showed how to learn chaotic dynamics along with the state trajectory from sequentially acquired observations, using local ensemble Kalman filters. Here, we more systematically investigate the possibility to use a local ensemble Kalman filter with either covariance localisation or local domains, in order to retrieve the state and a mix of key global and local parameters. Global parameters are meant to represent the surrogate dynamical core, for instance through a neural network, which is reminiscent of data-driven machine learning of dynamics, while the local parameters typically stand for the forcings of the model. Aiming at joint state and parameter estimation, a family of algorithms for covariance and local domain localisation is proposed. In particular, we show how to rigorously update global parameters using a local domain ensemble Kalman filter (EnKF) such as the local ensemble transform Kalman filter (LETKF), an inherently local method. The approach is tested with success on the 40-variable Lorenz model using several of the local EnKF flavors. A two-dimensional illustration based on a multi-layer Lorenz model is finally provided. It uses radiance-like non-local observations. It features both local domains and covariance localisation in order to learn the chaotic dynamics and the local forcings. This paper more generally addresses the key question of online estimation of both global and local model parameters.