# Learning Enhanced Ensemble Filters

arXiv: https://arxiv.org/abs/2504.17836 · HTML: https://mlsanity.com/paper/2504.17836/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 67 references that are in the map.
- First posted on arXiv 24 Apr 2025.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [CViT: Continuous Vision Transformer for Operator Learning](https://mlsanity.com/paper/2405.13998/) (2024)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [A Unified Perspective on the Dynamics of Deep Transformers](https://mlsanity.com/paper/2501.18322/) (2025)
- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [Data Assimilation: A Mathematical Introduction](https://mlsanity.com/paper/1506.07825/) (2015)
- [Ensemble Kalman Methods: A Mean Field Perspective](https://mlsanity.com/paper/2209.11371/) (2022)
- [Nonlinear Filtering with Brenier Optimal Transport Maps](https://mlsanity.com/paper/2310.13886/) (2023)
- [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)
- [Postprocessing of Ensemble Weather Forecasts Using Permutation-invariant Neural Networks](https://mlsanity.com/paper/2309.04452/) (2023)
- [Accurate deep learning-based filtering for chaotic dynamics by identifying instabilities without an ensemble](https://mlsanity.com/paper/2408.04739/) (2024)
- [Accuracy of the Ensemble Kalman Filter in the Near-Linear Setting](https://mlsanity.com/paper/2409.09800/) (2024)
- [BI-EqNO: Generalized Approximate Bayesian Inference with an Equivariant Neural Operator Framework](https://mlsanity.com/paper/2410.16420/) (2024)
- [Exact nonlinear state estimation](https://mlsanity.com/paper/2310.10976/) (2023)

## Built on by (most cited first, 11 of 11)

- [Flow Matching for Efficient and Scalable Data Assimilation](https://mlsanity.com/paper/2508.13313/) (2025)
- [One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators](https://mlsanity.com/paper/2605.06873/) (2026)
- [Closed-form conditional diffusion models for data assimilation](https://mlsanity.com/paper/2603.21291/) (2026)
- [Neural ensemble Kalman filter: Data assimilation for compressible flows with shocks](https://mlsanity.com/paper/2602.23461/) (2026)
- [Amortized Filtering and Smoothing with Conditional Normalizing Flows](https://mlsanity.com/paper/2604.07169/) (2026)
- [Operator Learning for Smoothing and Forecasting](https://mlsanity.com/paper/2603.20359/) (2026)
- [Auto-differentiable data assimilation: Co-learning of states, dynamics, and filtering algorithms](https://mlsanity.com/paper/2603.20891/) (2026)
- [Universal Approximation of Measure-to-Measure Operators by Pushforwards](https://mlsanity.com/paper/2609.35483/) (2026)
- [Attention Kernels for Learning Maps Between Heavy-Tailed Measures](https://mlsanity.com/paper/2610.00564/) (2026)
- [Conditional Normalizing Flows for Forward and Backward Joint State and Parameter Estimation](https://mlsanity.com/paper/2601.07013/) (2026)
- [Schur-Neural KF: Learned Schur-Consistent Corrections to the Extended Kalman Filter](https://mlsanity.com/paper/2609.32640/) (2026)

## Abstract (arXiv)

The filtering distribution in hidden Markov models evolves according to the law of a mean-field model in state-observation space. The ensemble Kalman filter (EnKF) approximates this mean-field model with an ensemble of interacting particles, employing a Gaussian ansatz for the joint distribution of the state and observation at each observation time. These methods are robust, but the Gaussian ansatz limits accuracy. Here this shortcoming is addressed by using machine learning to map the joint predicted state and observation to the updated state estimate. The derivation of methods from a mean field formulation of the true filtering distribution suggests a single parametrization of the algorithm that can be deployed at different ensemble sizes. And we use a mean field formulation of the ensemble Kalman filter as an inductive bias for our architecture. To develop this perspective, in which the mean-field limit of the algorithm and finite interacting ensemble particle approximations share a common set of parameters, a novel form of neural operator is introduced, taking probability distributions as input: a measure neural mapping (MNM). A MNM is used to design a novel approach to filtering, the MNM-enhanced ensemble filter (MNMEF), which is defined in both the mean-field limit and for interacting ensemble particle approximations. The ensemble approach uses empirical measures as input to the MNM and is implemented using the set transformer, which is invariant to ensemble permutation and allows for different ensemble sizes. In practice fine-tuning of a small number of parameters, for specific ensemble sizes, further enhances the accuracy of the scheme. The promise of the approach is demonstrated by its superior root-mean-square-error performance relative to leading methods in filtering the Lorenz '96 and Kuramoto-Sivashinsky models.