# The Ensemble Schr"odinger Bridge filter for Nonlinear Data Assimilation

arXiv: https://arxiv.org/abs/2512.18928 · HTML: https://mlsanity.com/paper/2512.18928/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 43 references that are in the map.
- First posted on arXiv 22 Dec 2025.

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems](https://mlsanity.com/paper/2309.00983/) (2023)
- [Contractive Diffusion Probabilistic Models](https://mlsanity.com/paper/2401.13115/) (2024)
- [Generative modeling for time series via Schr"odinger bridge](https://mlsanity.com/paper/2304.05093/) (2023)
- [Schr"odinger Bridge Samplers](https://mlsanity.com/paper/1912.13170/) (2019)
- [Ensemble score filter with image inpainting for data assimilation in tracking surface quasi-geostrophic dynamics with partial observations](https://mlsanity.com/paper/2501.12419/) (2025)
- [Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schroedinger bridge](https://mlsanity.com/paper/2005.10963/) (2020)

## Built on by (most cited first, 3 of 3)

- [SURGE: Approximation and Training Free Particle Filter for Diffusion Surrogate](https://mlsanity.com/paper/2605.18745/) (2026)
- [Pathwise Learning of Stochastic Dynamical Systems with Partial Observations](https://mlsanity.com/paper/2601.21860/) (2026)
- [Ensemble Controlled-Flow Filtering for Implicit Data Assimilation](https://mlsanity.com/paper/2607.12975/) (2026)

## Abstract (arXiv)

This work introduces a novel nonlinear optimal filtering method, termed the Ensemble Schr"odinger Bridge nonlinear filter. The proposed filter combines the standard prediction step with a diffusion-generative-modeling-based analysis step, thereby completing one full filtering update. The resulting approach introduces no structural model error, and is derivative-free, training-free, and highly parallelizable. Numerical experiments demonstrate that the proposed algorithm performs effectively for highly nonlinear dynamics and nonlinear observation processes, including chaotic systems with dimension up to 40 and beyond. The results also show that the method outperforms classical approaches such as the ensemble Kalman filter and particle filter across a range of tests with varying degrees of nonlinearity. Future work will focus on extending the proposed method to practical meteorological applications and developing a rigorous convergence theory.