# Ensemble Kalman filter in latent space using a variational autoencoder pair

arXiv: https://arxiv.org/abs/2502.12987 · HTML: https://mlsanity.com/paper/2502.12987/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 111 references that are in the map.
- First posted on arXiv 18 Feb 2025.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Diagnosing and Enhancing VAE Models](https://mlsanity.com/paper/1903.05789/) (2019)
- [Score-based Data Assimilation](https://mlsanity.com/paper/2306.10574/) (2023)
- [Taming VAEs](https://mlsanity.com/paper/1810.00597/) (2018)
- [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)
- [DiffDA: a Diffusion Model for Weather-scale Data Assimilation](https://mlsanity.com/paper/2401.05932/) (2024)
- [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)
- [Latent Space Data Assimilation by using Deep Learning](https://mlsanity.com/paper/2104.00430/) (2021)
- [Deep Generative Data Assimilation in Multimodal Setting](https://mlsanity.com/paper/2404.06665/) (2024)
- [On the Properties of Kullback-Leibler Divergence Between Multivariate Gaussian Distributions](https://mlsanity.com/paper/2102.05485/) (2021)
- [Latent-EnSF: A Latent Ensemble Score Filter for High-Dimensional Data Assimilation with Sparse Observation Data](https://mlsanity.com/paper/2409.00127/) (2024)
- [Inverse Problems and Data Assimilation](https://mlsanity.com/paper/1810.06191/) (2018)
- [Towards a Robust Parameterization for Conditioning Facies Models Using Deep Variational Autoencoders and Ensemble Smoother](https://mlsanity.com/paper/1812.06900/) (2018)
- [Online learning of both state and dynamics using ensemble Kalman filters](https://mlsanity.com/paper/2006.03859/) (2020)
- [Attention-based Convolutional Autoencoders for 3D-Variational Data Assimilation](https://mlsanity.com/paper/2101.02121/) (2021)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [Analog ensemble data assimilation and a method for constructing analogs with variational autoencoders](https://mlsanity.com/paper/2006.01101/) (2020)
- [Data Assimilation Networks](https://mlsanity.com/paper/2010.09694/) (2020)
- [Learning to Assimilate in Chaotic Dynamical Systems](https://mlsanity.com/paper/2111.01058/) (2021)
- [Learning Optimal Filters Using Variational Inference](https://mlsanity.com/paper/2406.18066/) (2024)
- [Accurate deep learning-based filtering for chaotic dynamics by identifying instabilities without an ensemble](https://mlsanity.com/paper/2408.04739/) (2024)
- [A comparison of nonlinear extensions to the ensemble Kalman filter: Gaussian Anamorphosis and Two-Step Ensemble Filters](https://mlsanity.com/paper/2107.07475/) (2021)
- [Towards diffusion models for large-scale sea-ice modelling](https://mlsanity.com/paper/2406.18417/) (2024)
- [AI-Aided Kalman Filters](https://mlsanity.com/paper/2410.12289/) (2024)

## Built on by (most cited first, 5 of 5)

- [Latent Twins](https://mlsanity.com/paper/2509.20615/) (2025)
- [ADDA: a Modular Framework for Representing, Simulating and Assimilating Dynamics with End-to-end Differentiability](https://mlsanity.com/paper/2608.23297/) (2026)
- [PnP-DA: Towards Principled Plug-and-Play Integration of Variational Data Assimilation and Generative Models](https://mlsanity.com/paper/2508.00325/) (2025)
- [Feature-preserving Latent-EnKF for Data Assimilation of Flows with Shocks](https://mlsanity.com/paper/2606.12559/) (2026)
- [Variational Augmented Invertible Koopman Autoencoder for probabilistic time series forecasting](https://mlsanity.com/paper/2609.37435/) (2026)

## Abstract (arXiv)

Popular (ensemble) Kalman filter data assimilation (DA) approaches assume that the errors in both the a priori estimate of the state and those in the observations are Gaussian. For constrained variables, e.g. sea ice concentration or stress, such an assumption does not hold. The variational autoencoder (VAE) is a machine learning (ML) technique that allows to map an arbitrary distribution to/from a latent space in which the distribution is supposedly closer to a Gaussian. We propose a novel hybrid DA-ML approach in which VAEs are incorporated in the DA procedure. Specifically, we introduce a variant of the popular ensemble transform Kalman filter (ETKF) in which the analysis is applied in the latent space of a single VAE or a pair of VAEs. In twin experiments with a simple circular model, whereby the circle represents an underlying submanifold to be respected, we find that the use of a VAE ensures that a posteri ensemble members lie close to the manifold containing the truth. Furthermore, online updating of the VAE is necessary and achievable when this manifold varies in time, i.e. when it is non-stationary. We demonstrate that introducing an additional second latent space for the observational innovations improves robustness against detrimental effects of non-Gaussianity and bias in the observational errors but it slightly lessens the performance if observational errors are strictly Gaussian.