Coupling-Informed Transport Maps for Bayesian Filtering in Nonlinear Dynamical Systems

2026-05-13

AI Summary

This work addresses the challenge of approximating posterior distributions in Bayesian filtering for nonlinear, non-Gaussian dynamic systems by proposing a training-free transport-based filtering method. Exploiting the coupling structure between states and observations, the approach constructs a block-triangular transport map that reformulates the analysis step as a maximum mean discrepancy (MMD) minimization problem. An analytical update direction is then obtained directly via gradient flow. To handle high-dimensional settings, a domain localization strategy is further incorporated. The method operates without particle sampling, thereby circumventing particle degeneracy, and demonstrates superior performance over conventional filters in numerical experiments while enjoying theoretical convergence guarantees.

Abstract

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.

Problem

• Bayesian filtering
• nonlinear dynamical systems
• non-Gaussian posteriors
• particle collapse
• transport maps

Innovation

• transport map
• maximum mean discrepancy
• likelihood-free filtering
• gradient flows
• non-Gaussian posterior