Training-Free Bayesian Filtering with Generative Emulators

This letter presents a unified formulation and a controlled numerical comparison of generative-model approaches to the nonlinear filtering problem. Under this formulation the analysis step is realized by a transport of the forecast distribution to the posterior, the approaches differing only in how that transport is selected and learned. We derive three new filters, based on stochastic interpolants, their deterministic flow-matching limit, and Schrödinger bridges realized through forward--backward SDEs. We develop a two-stage tuning procedure that separates the training of the generative model from its online refinement. The resulting methods are compared against the optimal transport filter (OTF), the Knothe--Rosenblatt filter (KRF), the sequential importance resampling (SIR) particle filter and the ensemble Kalman filter (EnKF), in terms of accuracy, computational time, and sensitivity to ensemble size and state dimension. The results indicate that every generative filter resolves multimodal posteriors that the EnKF and SIR do not, that no single generative framework dominates, the preferred method being set by the available online budget and ensemble size, and that the filters differ in the regularity of the particle trajectories they produce.

Data assimilation models state dynamics conditioned on sequential observations, and has wide-ranging scientific applications. In the filtering setting, the goal is to model the posterior over the current state given all observations so far. Classical solutions typically make simplifying distributional or functional assumptions, e.g., linear-Gaussian systems, which can be inaccurate in many scenarios. In principle, particle filters (PFs) remove these assumptions, yet often collapse in high dimensions. Recent generative approaches learn conditional state transitions, but without principled Bayesian updates they do not recover the correct filtering posterior and can accumulate error over long horizons. In this work, we introduce Flow Proposal Particle Filters (FPPF), which learn a conditional generative model based proposal approximating the variance-minimizing optimal proposal for particle propagation. Conditioning on observations steers particles toward high-likelihood regions before weighting, reducing weight variance and delaying degeneracy. Since our proposal admits tractable likelihood evaluation, FPPF computes accurate importance weights and retains a Bayesian update step. We further extend FPPF to high-dimensional problems through localization strategies, adressing another standard PF failure mode. Extensive experiments on a variety of dynamical systems show that FPPF outperforms statistical baselines and other generative methods in non-linear, non-Gaussian, and high-dimensional regimes.

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.