Bayesian Filtering in Physical Systems via Test-time Trained Flow Matching

Bayesian filtering provides a principled framework for online state estimation under uncertainty. Applying it to high-dimensional systems with complex posterior distributions remains challenging. Recent generative models, such as flow matching, have shown potential in Bayesian filtering. However, existing approaches either propagate a finite sample ensemble or repeatedly sample a state trajectory, coupling online cost to the ensemble size or trajectory sampling budget. To address this, we propose a new perspective of directly encoding the evolving distribution into flow matching model weights, namely, the Belief Flow Filter (BFF). It is a generative filtering framework that updates model weights via gradient descent at test time to track the posterior evolution. BFF carries the belief forward in generative model weights and draws samples from the updated model without restricting the posterior to a Gaussian family. A Bayes-derived loss motivates the update design, where an idealized expressivity result and a conditional, population-level bound connect the outer training loss to filtering error. The deployed finite-capacity update amortizes a specified observation model and has no guarantee of exact Bayesian inference under an unseen likelihood. Across five physical systems, BFF achieves the lowest reconstruction error and continuous ranked probability score (CRPS) on Kuramoto–Sivashinsky (KS) and Burgers' and ranks among the top two methods on all four metrics for Navier–Stokes (NS) flow with uniform sensors. BFF also achieves the lowest reconstruction error, spectrum error, and miscalibration area (MA) for NS flow with a single moving sensor, and the lowest gradient error, spectrum error, and MA for synthetic tokamak profile estimation from diagnostic measurements.