Bayesian Filtering in Physical Systems via Test-time Trained Flow Matching
Abstract
Bayesian filtering provides a principled framework for online state estimation under uncertainty, yet its application to systems with high-dimensional states and complicated posterior distributions remains challenging. Recent generative models, such as flow matching, have shown potential in Bayesian filtering. However, they still rely on particle-based representations of the posterior, which lose the rich information of the full distribution, or tackle a trajectory-level inverse problem that conflicts with the recursive structure of Bayesian filtering. 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. Thereby, BFF bypasses the scalability issue of particle representations or the flexibility limitation of Gaussian assumptions in conventional filters. We theoretically justify that the BFF design is structurally aligned with Bayesian filtering, and its training objective targets the recursive filtering operator. BFF is empirically verified across 5 different physical systems, including ones with chaotic dynamics and highly sparse, non-linear observations. The results show that BFF attains the best score in 8 of 9 metric-benchmark cells across the three standard 1D and 2D PDE benchmarks, and similarly leads on the extreme single-moving-sensor setting and on a real-world-grounded tokamak plasma estimation task, demonstrating its potential to accurately approximate the Bayesian filtering operator in high-dimensional probability space.
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BibTeX
@misc{feng2026bayesian,
title = {{Bayesian Filtering in Physical Systems via Test-time Trained Flow Matching}},
author = {Ruiqi Feng and Chongyi Wang and Tao Zhang and Tailin Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23383},
eprint = {2609.23383},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23383},
}
RIS
TY - UNPB
TI - Bayesian Filtering in Physical Systems via Test-time Trained Flow Matching
AU - Ruiqi Feng
AU - Chongyi Wang
AU - Tao Zhang
AU - Tailin Wu
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23383
UR - https://arxiv.org/abs/2609.23383
AB - Bayesian filtering provides a principled framework for online state estimation under uncertainty, yet its application to systems with high-dimensional states and complicated posterior distributions remains challenging. Recent generative models, such as flow matching, have shown potential in Bayesian filtering. However, they still rely on particle-based representations of the posterior, which lose the rich information of the full distribution, or tackle a trajectory-level inverse problem that conflicts with the recursive structure of Bayesian filtering. 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. Thereby, BFF bypasses the scalability issue of particle representations or the flexibility limitation of Gaussian assumptions in conventional filters. We theoretically justify that the BFF design is structurally aligned with Bayesian filtering, and its training objective targets the recursive filtering operator. BFF is empirically verified across 5 different physical systems, including ones with chaotic dynamics and highly sparse, non-linear observations. The results show that BFF attains the best score in 8 of 9 metric-benchmark cells across the three standard 1D and 2D PDE benchmarks, and similarly leads on the extreme single-moving-sensor setting and on a real-world-grounded tokamak plasma estimation task, demonstrating its potential to accurately approximate the Bayesian filtering operator in high-dimensional probability space.
ER -