# Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems

arXiv: https://arxiv.org/abs/2412.14318 · HTML: https://mlsanity.com/paper/2412.14318/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 77 references that are in the map.
- First posted on arXiv 18 Dec 2024.

## Builds on (most cited first, 30 of 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [ClimaX: A foundation model for weather and climate](https://mlsanity.com/paper/2301.10343/) (2023)
- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [FengWu: Pushing the Skillful Global Medium-range Weather Forecast beyond 10 Days Lead](https://mlsanity.com/paper/2304.02948/) (2023)
- [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)
- [Training neural operators to preserve invariant measures of chaotic attractors](https://mlsanity.com/paper/2306.01187/) (2023)
- [Using machine learning to correct model error in data assimilation and forecast applications](https://mlsanity.com/paper/2010.12605/) (2020)
- [Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization](https://mlsanity.com/paper/2001.06270/) (2020)
- [Exact minimax risk for linear least squares, and the lower tail of sample covariance matrices](https://mlsanity.com/paper/1912.10754/) (2019)
- [Auto-differentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2107.07687/) (2021)
- [Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation](https://mlsanity.com/paper/2007.07383/) (2020)
- [FengWu-4DVar: Coupling the Data-driven Weather Forecasting Model with 4D Variational Assimilation](https://mlsanity.com/paper/2312.12455/) (2023)
- [Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations](https://mlsanity.com/paper/2108.03561/) (2021)
- [Integrating Recurrent Neural Networks with Data Assimilation for Scalable Data-Driven State Estimation](https://mlsanity.com/paper/2109.12269/) (2021)
- [Ensemble Kalman Methods: A Mean Field Perspective](https://mlsanity.com/paper/2209.11371/) (2022)
- [Towards physically consistent data-driven weather forecasting: Integrating data assimilation with equivariance-preserving deep spatial transformers](https://mlsanity.com/paper/2103.09360/) (2021)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Deep learning-enhanced ensemble-based data assimilation for high-dimensional nonlinear dynamical systems](https://mlsanity.com/paper/2206.04811/) (2022)
- [EM-like Learning Chaotic Dynamics from Noisy and Partial Observations](https://mlsanity.com/paper/1903.10335/) (2019)
- [Data Assimilation with Machine Learning Surrogate Models: A Case Study with FourCastNet](https://mlsanity.com/paper/2405.13180/) (2024)
- [Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization](https://mlsanity.com/paper/2208.03246/) (2022)
- [Machine Learning for Inverse Problems and Data Assimilation](https://mlsanity.com/paper/2410.10523/) (2024)
- [The Mean Field Ensemble Kalman Filter: Near-Gaussian Setting](https://mlsanity.com/paper/2212.13239/) (2022)
- [Accuracy of the Ensemble Kalman Filter in the Near-Linear Setting](https://mlsanity.com/paper/2409.09800/) (2024)
- [Learning Dissipative Chaotic Dynamics with Boundedness Guarantees](https://mlsanity.com/paper/2410.00976/) (2024)
- [Kinetic Methods for Inverse Problems](https://mlsanity.com/paper/1811.09387/) (2018)
- [Exploring the Use of Machine Learning Weather Models in Data Assimilation](https://mlsanity.com/paper/2411.14677/) (2024)
- [Ensemble Kalman Filters with Resampling](https://mlsanity.com/paper/2308.08751/) (2023)

## Built on by (most cited first, 1 of 1)

- [Filtering with Randomised Observations: Sequential Learning of Relevant Subspace Properties and Accuracy Analysis](https://mlsanity.com/paper/2509.04867/) (2025)

## Abstract (arXiv)

Filtering is concerned with online estimation of the state of a dynamical system from partial and noisy observations. In applications where the state is high dimensional, ensemble Kalman filters are often the method of choice. This paper establishes long-time accuracy of ensemble Kalman filters. We introduce conditions on the dynamics and the observations under which the estimation error remains small in the long-time horizon. Our theory covers a wide class of partially-observed chaotic dynamical systems, which includes the Navier-Stokes equations and Lorenz models. In addition, we prove long-time accuracy of ensemble Kalman filters with surrogate dynamics, thus validating the use of machine-learned forecast models in ensemble data assimilation.