Bayesian filtering improves physical system tracking using test-time learning
Bayesian Filtering in Physical Systems via Test-time Trained Flow Matching
Artificial IntelligenceMachine Learning
Summary
Tracking the state of complex physical systems in real-time is hard, especially when the system’s internal state is very complicated. The authors introduce the Belief Flow Filter (BFF), a new method that updates how a model understands the system’s state as it receives new data, by adjusting the model itself during tracking. This approach avoids common problems of older methods that struggle with complex states or rely on simpler assumptions. Tests on various physical systems, including those with chaotic behaviors and limited observations, show BFF can track states more accurately than existing methods.
What this means in practice
- •For control engineers: Improve real-time monitoring and control by accurately estimating complex system states in devices like tokamak plasma reactors.
- •For sensor system designers: Enhance state estimation when using very limited or sparse sensor data in high-dimensional physical systems.
Authors
Ruiqi Feng, Chongyi Wang, Tao Zhang, Tailin Wu
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.