Learning Enhanced Ensemble Filters

2025-04-24

AI Summary

Traditional ensemble Kalman filtering (EnKF) for hidden Markov models suffers from accuracy degradation in non-Gaussian settings due to its restrictive Gaussian assumption. To address this, we propose a novel non-Gaussian ensemble filtering framework grounded in mean-field evolution. Our key contribution is the first design of a Measure Neural Mapping (MNM) operator that takes empirical measures as input and—coupled with the deep set-invariant architecture Set Transformer—enables parameter sharing and scale-robust modeling over variable-size ensembles. Unlike EnKF, our method dispenses with the joint Gaussianity assumption and directly learns nonlinear probabilistic mappings between state and observation spaces. Experiments on the Lorenz-96 and Kuramoto–Sivashinsky systems demonstrate that our approach achieves significantly lower root-mean-square error (RMSE) than EnKF, particle filters, and other mainstream methods, while exhibiting superior accuracy and generalization across diverse dynamical regimes.

Technology Category

• Machine Learning: Ensemble Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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. This shortcoming is addressed by approximating the mean-field evolution using a novel form of neural operator 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-fieldlimit 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. The derivation of methods from a mean-field formulation allows a single parameterization of the algorithm to be deployed at 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.

Problem

• Overcoming Gaussian ansatz limitations in ensemble Kalman filters
• Designing neural operator-based filters for mean-field evolution
• Enhancing accuracy in filtering nonlinear dynamical systems

Innovation

• Uses Measure Neural Mapping for mean-field evolution
• Implements ensemble filter with set transformer
• Fine-tunes parameters for enhanced accuracy