Shiwei Ni | Searcharxiv

arXiv2026

FREESIA: Covariance-Aware Posterior Transport for Expressive and Scalable Data Assimilation

Data assimilation aims to infer the state of complex dynamical systems based on observational data. However, accurate inference of the multimodal posteriors induced by nonlinear or non-injective observation operators remains a key challenge under high-dimensional and sparse observation conditions. Ensemble filters scale to high dimensions but are confined by restrictive distributional assumptions, while training-free generative filters (e.g., EnSF, EnFF) alleviate this limitation but may introduce structural errors and hinder information propagation under sparse observations. To address these issues, we propose a training-free, asymptotically exact posterior transport method. Firstly, a covariance-aware posterior transport scheme is designed, which embeds the forecast cross-covariance into flow-based transport and accurately recovers unobserved states while preserving the non-Gaussian posterior structure. Furthermore, the method combines a tractable observation-adaptive proposal with posterior correction, ensuring accurate approximation of the nonlinear posterior distribution. Finally, we establish the corresponding posterior flow theory, from which the asymptotic exactness of the proposed method relative to finite-ensemble surrogates and the Wasserstein error bound are derived. Experiments on Double-Well, Lorenz-96, and Kolmogorov flow show that the proposed method captures complex posterior structure and remains accurate under sparse, nonlinear, and non-injective observations. In the sparse non-injective setting, it reduces RMSE by 56% relative to the best baseline.

stat.ML↗

arXiv2026

AECSF: Adaptive Ensemble Conditional Score Filtering for High-Dimensional Nonlinear Data Assimilation

Bayesian state estimation for high-dimensional nonlinear dynamical systems entails a fundamental tension between statistical fidelity and computational tractability, as particle weights can collapse, while Gaussian ensemble updates can miss non-Gaussian posterior structure. Score-based diffusion filters offer a sampling-based alternative, but existing training-free score filters often rely on heuristic likelihood corrections, which can compromise posterior accuracy by neglecting uncertainty about the system state associated with each noisy reverse particle. To address these issues, we propose AECSF, a training-free adaptive ensemble conditional score filter. AECSF constructs an analytically tractable score estimator from the conditional Tweedie identity, which recasts noisy posterior score estimation as estimating the conditional mean of the system state given a noisy reverse particle and the observation. To estimate these conditional means efficiently, AECSF employs a shared adaptive weighted proposal ensemble, while particle-specific conditional weights yield an estimate for each noisy reverse particle without separate proposal sampling. The proposal ensemble is updated using reverse-particle information within the same reverse-diffusion run to improve conditional-mean estimation. Theoretically, we characterize when a fixed weighted proposal measure yields the exact noisy posterior score. Under stated assumptions, we establish a bound relating conditional-mean estimation errors to reverse-sampling endpoint error. Numerical experiments demonstrate that AECSF improves the accuracy of posterior sampling and nonlinear filtering in high-dimensional problems with limited forecast ensembles.

stat.ML↗