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Ensemble Transport Filter via Optimized Maximum Mean Discrepancy

Dengfei Zeng, Lijian Jiang

TL;DR

The paper tackles nonlinear data assimilation in high-dimensional, non-Gaussian settings where traditional EnKF struggles and particle filters suffer from degeneracy. It introduces the Ensemble Transport Filter (EnTranF), which learns a transport map from the forecast prior to the posterior by minimizing a Maximum Mean Discrepancy ($MMD$) loss in an RKHS, and augments this with a variance penalty to emphasize informative statistics. A key advancement is the high-dimensional extension via observation splitting, enabling sequential, lower-dimensional Bayesian inversions that are composed to form the full update. The method is analyzed shows that in the linear-kernel limit it recovers an EnKF-like update, while in nonlinear settings it yields non-Gaussian posterior fidelity, demonstrated across static inverse problems and chaotic Lorenz systems. Overall, EnTranF provides a robust, scalable alternative to EnKF and PF for complex assimilation tasks, with demonstrated improvements in posterior accuracy, uncertainty quantification, and stability.

Abstract

In this paper, we present a new ensemble-based filter method by reconstructing the analysis step of the particle filter through a transport map, which directly transports prior particles to posterior particles. The transport map is constructed through an optimization problem described by the Maximum Mean Discrepancy loss function, which matches the expectation information of the approximated posterior and reference posterior. The proposed method inherits the accurate estimation of the posterior distribution from particle filtering while gives an extension to high dimensional assimilation problems. To improve the robustness of Maximum Mean Discrepancy, a variance penalty term is used to guide the optimization. It prioritizes minimizing the discrepancy between the expectations of highly informative statistics for the reference posteriors. The penalty term significantly enhances the robustness of the proposed method and leads to a better approximation of the posterior. A few numerical examples are presented to illustrate the advantage of the proposed method over ensemble Kalman filter.

Ensemble Transport Filter via Optimized Maximum Mean Discrepancy

TL;DR

The paper tackles nonlinear data assimilation in high-dimensional, non-Gaussian settings where traditional EnKF struggles and particle filters suffer from degeneracy. It introduces the Ensemble Transport Filter (EnTranF), which learns a transport map from the forecast prior to the posterior by minimizing a Maximum Mean Discrepancy () loss in an RKHS, and augments this with a variance penalty to emphasize informative statistics. A key advancement is the high-dimensional extension via observation splitting, enabling sequential, lower-dimensional Bayesian inversions that are composed to form the full update. The method is analyzed shows that in the linear-kernel limit it recovers an EnKF-like update, while in nonlinear settings it yields non-Gaussian posterior fidelity, demonstrated across static inverse problems and chaotic Lorenz systems. Overall, EnTranF provides a robust, scalable alternative to EnKF and PF for complex assimilation tasks, with demonstrated improvements in posterior accuracy, uncertainty quantification, and stability.

Abstract

In this paper, we present a new ensemble-based filter method by reconstructing the analysis step of the particle filter through a transport map, which directly transports prior particles to posterior particles. The transport map is constructed through an optimization problem described by the Maximum Mean Discrepancy loss function, which matches the expectation information of the approximated posterior and reference posterior. The proposed method inherits the accurate estimation of the posterior distribution from particle filtering while gives an extension to high dimensional assimilation problems. To improve the robustness of Maximum Mean Discrepancy, a variance penalty term is used to guide the optimization. It prioritizes minimizing the discrepancy between the expectations of highly informative statistics for the reference posteriors. The penalty term significantly enhances the robustness of the proposed method and leads to a better approximation of the posterior. A few numerical examples are presented to illustrate the advantage of the proposed method over ensemble Kalman filter.
Paper Structure (19 sections, 4 theorems, 55 equations, 20 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 4 theorems, 55 equations, 20 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Bayesian formulation of nonlinear filter
  3. State estimation of dynamical system
  4. Sequential Bayesian Filtering
  5. Ensemble Transport Filter
  6. Proposal transition generated by transport map in analysis step
  7. Ensemble approximation of transport filter
  8. Variance penalty for MMD loss
  9. An extension to high dimensional problems
  10. Connection with Ensemble Kalman Filter
  11. Numerical Results
  12. Static inverse problem
  13. Double-Well potential
  14. Lorenz'63 System
  15. Lorenz'96 System
  16. ...and 4 more sections

Key Result

Lemma 3.1

(Lemma 9.3.2 in richardmRealAnalysisProbability2018) Let $(\Omega, d)$ be a metric space, and $p, q$ be two probability measure defined on $\Omega$. Then $p=q$ if and only if $\mathbb{E}_{\boldsymbol{x}\sim p}[f(\boldsymbol{x})] = \mathbb{E}_{\boldsymbol{y}\sim q}[f(\boldsymbol{y})]$ for all $f\in \

Figures (20)

  • Figure 2.1: Directed probability graph of dynamical model \ref{['eq:dynamical model']} and observation operator \ref{['eq:observation operator']} in the form of state-space model
  • Figure 2.2: Propagation of particles between distributions in ensemble-based filtering
  • Figure 3.3: Schematic of observation split and composition of transport maps generated by each splited observation.
  • Figure 4.4: Posterior approximation of EnTranF with Linear transport map (Panel (a)-(c)) and Nonlinear transport map (Panel (d)-(f)) when taking different bandwidths ($10^{-3}$, $10^{0}$ and $10^{3}$).
  • Figure 4.5: 2-d Histogram of ensemble particles generated by EnKF, EnTranF and EnTranFp. Red "$\times$" represents the MAP estimation of the corresponding method.
  • ...and 15 more figures

Theorems & Definitions (9)

  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.4
  • proof
  • Remark 3.3
  • Remark 3.4