Physics-informed neural particle flow for the Bayesian update step

Physics-informed neural particle flow (PINPF) learns a deterministic transport field that moves particles from a prior distribution toward a Bayesian posterior while enforcing the governing probability-evolution equation. However, the standard PINPF velocity model processes particles independently and therefore does not explicitly condition its transport decisions on the empirical particle population. This paper introduces population-aware PINPF (PA-PINPF), which augments each particle update with a permutation-invariant Deep Sets representation of the full particle set. We investigate two population encoders. PA-PINPF-State summarizes the particle states, whereas PA-PINPF-Feature summarizes the complete local physics-informed feature vectors, including particle position, pseudo-time, measurement information, likelihood values, and score information. The latter allows the population context to represent not only particle-cloud geometry, but also the population-level Bayesian transport geometry. The methods retain the original unsupervised physics-informed residual objective and require no ground-truth posterior samples during training. Experiments on range-measurement tasks and nonlinear time-difference-of-arrival posterior transport demonstrate that both population-aware variants improve over particle-wise PINPF, while feature-population encoding provides the strongest performance. These results show that population-level physics features provide useful global information for learned Bayesian particle transport.

Sequential inference estimates latent states from noisy and incomplete observations. Particle Filters (PFs), a class of Monte Carlo methods based on importance sampling, provide a flexible framework for this task, but often suffer from poor sample efficiency and unfavorable scaling with dimension, partly due to suboptimal proposal distributions. We address these challenges by integrating learned proposals into the PF framework. We introduce Neural Optimal Particle Filters (NOPFs), which learn an amortized approximation to the optimal proposal from offline simulated one-step conditioning tuples. The learned proposal is used as a drop-in replacement in standard PF updates, with samples corrected by standard importance weights so that the method asymptotically targets the same filtering distribution under standard support and density-evaluation assumptions. Across stochastic nonlinear benchmarks of varying inference complexity, NOPFs improve sample efficiency and distributional accuracy over standard PF baselines with modest computational overhead. The approach integrates data-driven proposal learning into classical inference without altering the underlying filtering objective.

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.