The Ensemble Schr"odinger Bridge filter for Nonlinear Data Assimilation

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.