Data assimilation (DA) integrates observations with a dynamical model to estimate states of PDE-governed systems. Model-driven methods (e.g., Kalman Filter, Particle Filter) presuppose full knowledge of the true dynamics, which is not always satisfied in practice, while purely data-driven solvers learn a deterministic mapping between observations and states and therefore miss the intrinsic stochasticity of real processes. Recently, score-based diffusion models have shown promise for DA by learning a global diffusion prior to represent stochastic dynamics. However, their one-shot generation lacks stepwise physical consistency and struggles with complex stochastic processes. To address these issues, we propose FlowDAS, a generative DA framework that employs stochastic interpolants to learn state transition dynamics through step-by-step stochastic updates. By incorporating observations into each transition, FlowDAS can produce stable, measurement-consistent forecasts.

Data assimilation (DA) addresses the problem of sequentially estimating the state of a dynamical system from noisy and incomplete observations. In this work, we employ a diffusion model as a world model to simulate and predict the system's dynamics. Recently, score-based diffusion models have learned global diffusion priors that effectively model (stochastic) dynamics, revealing strong potential for data assimilation. In this paper, we investigate how information from noisy observations can be incorporated to enable continuous correction and refinement of the predicted system state when using a diffusion prior. Motivated by particle filtering methods, we represent the posterior distribution using a set of particles. After receiving noisy observations, the diffusion model is guided using the observation likelihood to steer the generation process toward observation-consistent states. Nevertheless, such guidance does not guarantee sampling from the true posterior. We therefore employ a Sequential Monte Carlo approach over the diffusion trajectory, viewed as a path measure, to reweight and resample particles, thereby correcting the generation process and ensuring convergence toward the desired posterior distribution. This leads to an unbiased particle filtering method that rigorously fuses observational data with diffusion model simulations.

Data assimilation (DA) is crucial for improving the accuracy of state estimation in complex dynamical systems by integrating observational data with physical models. Traditional solutions rely on either pure model-driven approaches, such as Bayesian filters that struggle with nonlinearity, or data-driven methods using deep learning priors, which often lack generalizability and physical interpretability. Recently, score-based DA methods have been introduced, focusing on learning prior distributions but neglecting explicit state transition dynamics, leading to limited accuracy improvements. To tackle the challenge, we introduce FlowDAS, a novel generative model-based framework using the stochastic interpolants to unify the learning of state transition dynamics and generative priors. FlowDAS achieves stable and observation-consistent inference by initializing from proximal previous states, mitigating the instability seen in score-based methods. Our extensive experiments demonstrate FlowDAS's superior performance on various benchmarks, from the Lorenz system to high-dimensional fluid super-resolution tasks. FlowDAS also demonstrates improved tracking accuracy on practical Particle Image Velocimetry (PIV) task, showcasing its effectiveness in complex flow field reconstruction.