# Nonlinear Filtering with Brenier Optimal Transport Maps

arXiv: https://arxiv.org/abs/2310.13886 · HTML: https://mlsanity.com/paper/2310.13886/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 52 references that are in the map.
- First posted on arXiv 21 Oct 2023.

## Builds on (most cited first, 18 of 18)

- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [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)
- [Preconditioned training of normalizing flows for variational inference in inverse problems](https://mlsanity.com/paper/2101.03709/) (2021)
- [Ensemble Kalman Methods: A Mean Field Perspective](https://mlsanity.com/paper/2209.11371/) (2022)
- [The efficacy and generalizability of conditional GANs for posterior inference in physics-based inverse problems](https://mlsanity.com/paper/2202.07773/) (2022)
- [An Optimal Transport Formulation of Bayes' Law for Nonlinear Filtering Algorithms](https://mlsanity.com/paper/2203.11869/) (2022)
- [McKean-Vlasov SDEs in nonlinear filtering](https://mlsanity.com/paper/2007.12658/) (2020)
- [Bayesian Posterior Perturbation Analysis with Integral Probability Metrics](https://mlsanity.com/paper/2303.01512/) (2023)
- [Optimal Transport Particle Filters](https://mlsanity.com/paper/2304.00392/) (2023)
- [A Survey of Feedback Particle Filter and related Controlled Interacting Particle Systems (CIPS)](https://mlsanity.com/paper/2301.00935/) (2023)
- [Computational Optimal Transport and Filtering on Riemannian manifolds](https://mlsanity.com/paper/2309.08847/) (2023)
- [Stochastic Particle Flow for Nonlinear High-Dimensional Filtering Problems](https://mlsanity.com/paper/1511.01448/) (2015)

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

- [Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior](https://mlsanity.com/paper/2406.07475/) (2024)
- [Learning Enhanced Ensemble Filters](https://mlsanity.com/paper/2504.17836/) (2025)
- [Learning Probabilistic Filters with Strictly Proper Scoring Rules](https://mlsanity.com/paper/2606.26497/) (2026)
- [Fast filtering of non-Gaussian models using Amortized Optimal Transport Maps](https://mlsanity.com/paper/2503.12633/) (2025)
- [Closed-form conditional diffusion models for data assimilation](https://mlsanity.com/paper/2603.21291/) (2026)
- [Ensemble Transport Filter via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/2407.11518/) (2024)
- [Conditional Sampling via Wasserstein Autoencoders and Triangular Transport](https://mlsanity.com/paper/2604.02644/) (2026)
- [Preserving linear invariants in ensemble filtering methods](https://mlsanity.com/paper/2404.14328/) (2024)
- [Differentiable Adaptive Kalman Filtering via Optimal Transport](https://mlsanity.com/paper/2508.07037/) (2025)
- [Coupling-Informed Transport Maps for Bayesian Filtering in Nonlinear Dynamical Systems](https://mlsanity.com/paper/2605.13174/) (2026)
- [Generative models for simulation based filtering: Formulations and Empirical Comparisons](https://mlsanity.com/paper/2609.16317/) (2026)

## Abstract (arXiv)

This paper is concerned with the problem of nonlinear filtering, i.e., computing the conditional distribution of the state of a stochastic dynamical system given a history of noisy partial observations. Conventional sequential importance resampling (SIR) particle filters suffer from fundamental limitations, in scenarios involving degenerate likelihoods or high-dimensional states, due to the weight degeneracy issue. In this paper, we explore an alternative method, which is based on estimating the Brenier optimal transport (OT) map from the current prior distribution of the state to the posterior distribution at the next time step. Unlike SIR particle filters, the OT formulation does not require the analytical form of the likelihood. Moreover, it allows us to harness the approximation power of neural networks to model complex and multi-modal distributions and employ stochastic optimization algorithms to enhance scalability. Extensive numerical experiments are presented that compare the OT method to the SIR particle filter and the ensemble Kalman filter, evaluating the performance in terms of sample efficiency, high-dimensional scalability, and the ability to capture complex and multi-modal distributions.