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This paper tackles the intricate task of jointly estimating state andparameters in data assimilation for stochastic dynamical systems that areaffected by noise and observed only partially. While the concept of ``optimalfiltering'' serves as the customary approach to estimate the state of thetarget dynamical system, traditional methods such as Kalman filters andparticle filters encounter significant challenges when dealing withhigh-dimensional and nonlinear problems. When we also consider the scenariowhere the model parameters are unknown, the problem transforms into a jointstate-parameter estimation endeavor. Presently, the leading-edge techniqueknown as the Augmented Ensemble Kalman Filter (AugEnKF) addresses this issue bytreating unknown parameters as additional state variables and employing theEnsemble Kalman Filter to estimate the augmented state-parameter vector.Despite its considerable progress, AugEnKF does exhibit certain limitations interms of accuracy and stability. To address these challenges, we introduce aninnovative approach, referred to as the United Filter. This method amalgamatesa remarkably stable and efficient ensemble score filter (EnSF) for stateestimation with a precise direct filter dedicated to online parameterestimation. Utilizing the EnSF's generative capabilities grounded in diffusionmodels, the United Filter iteratively fine-tunes both state and parameterestimates within a single temporal data assimilation step. Thanks to therobustness of the EnSF, the proposed method offers a promising solution forenhancing our comprehension and modeling of dynamical systems, as demonstratedby results from numerical experiments.

A Unified Filter Method for Jointly Estimating State and Parameters of Stochastic Dynamical Systems via the Ensemble Score Filter