# Ensemble Kalman Methods: A Mean Field Perspective

arXiv: https://arxiv.org/abs/2209.11371 · HTML: https://mlsanity.com/paper/2209.11371/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2022.

## Built on by (most cited first, 20 of 20)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Ensemble Kalman Diffusion Guidance: A Derivative-free Method for Inverse Problems](https://mlsanity.com/paper/2409.20175/) (2024)
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- [Nonlinear Filtering with Brenier Optimal Transport Maps](https://mlsanity.com/paper/2310.13886/) (2023)
- [Learning Enhanced Ensemble Filters](https://mlsanity.com/paper/2504.17836/) (2025)
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- [Learning Optimal Filters Using Variational Inference](https://mlsanity.com/paper/2406.18066/) (2024)
- [Hyperparameter Optimization for Randomized Algorithms: A Case Study on Random Features](https://mlsanity.com/paper/2407.00584/) (2024)
- [Exact nonlinear state estimation](https://mlsanity.com/paper/2310.10976/) (2023)
- [One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators](https://mlsanity.com/paper/2605.06873/) (2026)
- [Generative Modelling with Tensor Train approximations of Hamilton--Jacobi--Bellman equations](https://mlsanity.com/paper/2402.15285/) (2024)
- [Bayesian Experimental Design for Model Discrepancy Calibration: An Auto-Differentiable Ensemble Kalman Inversion Approach](https://mlsanity.com/paper/2504.20319/) (2025)
- [Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems](https://mlsanity.com/paper/2412.14318/) (2024)
- [Nesterov Acceleration for Ensemble Kalman Inversion and Variants](https://mlsanity.com/paper/2501.08779/) (2025)
- [Ensemble Kalman-Bucy filtering for nonlinear model predictive control](https://mlsanity.com/paper/2503.12474/) (2025)
- [Gradient-Free Score-Based Sampling Methods with Ensembles](https://mlsanity.com/paper/2401.17539/) (2024)
- [Bayesian Experimental Design for Model Discrepancy Calibration: A Rivalry between Kullback--Leibler Divergence and Wasserstein Distance](https://mlsanity.com/paper/2601.16425/) (2026)
- [A Continuous-Time Ensemble Kalman-Bucy Smoother for Causal Inference and Model Discovery](https://mlsanity.com/paper/2604.25157/) (2026)
- [Calibrating subgrid parametrizations of single-column ocean models via simulation-based inference](https://mlsanity.com/paper/2609.13242/) (2026)

## Abstract (arXiv)

Ensemble Kalman methods are widely used for state estimation in the geophysical sciences. Their success stems from the fact that they take an underlying (possibly noisy) dynamical system as a black box to provide a systematic, derivative-free methodology for incorporating noisy, partial and possibly indirect observations to update estimates of the state; furthermore the ensemble approach allows for sensitivities and uncertainties to be calculated. The methodology was introduced in 1994 in the context of ocean state estimation. Soon thereafter it was adopted by the numerical weather prediction community and is now a key component of the best weather prediction systems worldwide. Furthermore the methodology is starting to be widely adopted for numerous problems in the geophysical sciences and is being developed as the basis for general purpose derivative-free inversion methods that show great promise. Despite this empirical success, analysis of the accuracy of ensemble Kalman methods, in terms of their capabilities as both state estimators and quantifiers of uncertainty, is lagging. The purpose of this paper is to provide a unifying mean field based framework for the derivation and analysis of ensemble Kalman methods. Both state estimation and parameter estimation problems (inverse problems) are considered, and formulations in both discrete and continuous time are employed. For state estimation problems, both the control and filtering approaches are considered; analogously for parameter estimation problems, the optimization and Bayesian perspectives are both studied. The mean field perspective provides an elegant framework, suitable for analysis; furthermore, a variety of methods used in practice can be derived from mean field systems by using interacting particle system approximations. The approach taken also unifies a wide-ranging literature in the field and suggests open problems.