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

TL;DR

The paper introduces the Belief Flow Filter (BFF), a generative filtering method that updates flow matching model weights during inference to track evolving posterior distributions in physical systems. By directly encoding belief states into model parameters, BFF avoids particle approximations and trajectory-level inverse problems, enabling accurate, scalable Bayesian filtering in high-dimensional, nonlinear settings.

Why It Matters

BFF enables robust state estimation in complex physical systems with sparse or chaotic dynamics, where traditional filters fail due to assumptions about distribution shape or scalability. This approach advances real-world applications like plasma control in fusion energy, where precise, real-time uncertainty quantification is critical.

Key Findings

• BFF updates flow matching model weights at test time to represent the posterior distribution, eliminating reliance on particle representations.
• The method aligns with the recursive structure of Bayesian filtering and achieves state-of-the-art performance across multiple PDE benchmarks and a tokamak plasma estimation task.
• BFF outperforms existing methods in 8 out of 9 evaluation metrics on standard benchmarks and excels in extreme sensing scenarios with limited observations.