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

arXiv: https://arxiv.org/abs/2605.13174 · HTML: https://mlsanity.com/paper/2605.13174/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 35 references that are in the map.
- First posted on arXiv 13 May 2026.

## Builds on (most cited first, 10 of 10)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [The Ensemble Kalman Filter: A Signal Processing Perspective](https://mlsanity.com/paper/1702.08061/) (2017)
- [Nonlinear Filtering with Brenier Optimal Transport Maps](https://mlsanity.com/paper/2310.13886/) (2023)
- [Diffusion map-based algorithm for Gain function approximation in the Feedback Particle Filter](https://mlsanity.com/paper/1902.07263/) (2019)
- [Deep FPF: Gain function approximation in high-dimensional setting](https://mlsanity.com/paper/2010.01183/) (2020)
- [Ensemble Transport Filter via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/2407.11518/) (2024)

## Abstract (arXiv)

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.