The Ensemble Schr{\"o}dinger Bridge filter for Nonlinear Data Assimilation

TL;DR

This paper presents a new nonlinear filtering method called the Ensemble Schrödinger Bridge filter, which effectively handles highly nonlinear and chaotic systems without structural model error.

Contribution

It introduces a derivative-free, training-free, and parallelizable filtering approach combining diffusion models with ensemble methods, outperforming classical filters.

Findings

Effective for chaotic systems with high dimensions.

Outperforms ensemble Kalman and particle filters.

No structural model error and highly parallelizable.

Abstract

This work introduces a novel nonlinear optimal filtering method, termed the Ensemble Schr{\"o}dinger 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.