Physics-informed neural particle flow for the Bayesian update step

TL;DR

This work proposes a physics-informed neural particle flow, which is an amortized inference framework that trains a neural network to approximate the transport velocity field, and demonstrates that the neural parameterization acts as an implicit regularizer, mitigating the numerical stiffness inherent to analytic flows and reducing online computational complexity.

Abstract

The Bayesian update step poses significant computational challenges in high-dimensional nonlinear estimation. While log-homotopy particle flow filters offer an alternative to stochastic sampling, existing formulations usually yield stiff differential equations. Conversely, existing deep learning approximations typically treat the update as a black-box task or rely on asymptotic relaxation, neglecting the exact geometric structure of the finite-horizon probability transport. In this work, we propose a physics-informed neural particle flow, which is an amortized inference framework. To construct the flow, we couple the log-homotopy trajectory of the prior to posterior density function with the continuity equation describing the density evolution. This derivation yields a governing partial differential equation (PDE), referred to as the master PDE. By embedding this PDE as a physical constraint into the loss function, we train a neural network to approximate the transport velocity field. This approach enables purely unsupervised training, eliminating the need for ground-truth posterior samples. We demonstrate that the neural parameterization acts as an implicit regularizer, mitigating the numerical stiffness inherent to analytic flows and reducing online computational complexity. Experimental validation on multimodal benchmarks and a challenging nonlinear scenario confirms better mode coverage and robustness compared to state-of-the-art baselines.

Figures (10)

• Figure 1: Physics based Bayesian computation. Center: The estimation objectiveis a static target posterior and its discrete approximation via an empirical measure. Left: Stochastic relaxation, where the system asymptotically converges to a fluctuating equilibrium ($t \to \infty$). The macroscopic and microscopic views are linked via the mean-field limit. Right: Deterministic transport, where the system evolves over a finite horizon ($\lambda \in [0,1]$). The particle trajectories correspond to the characteristic curves of the macroscopic master PDE.
• Figure 2: Corner plot for a representative test inference task, which is selected as the one whose quantitative performance is closest to the mean across all 100 test problems (Table \ref{['tab:4dgmm-general']}). Posterior samples from our method, SVGD, and annealed MCMC are shown alongside the analytic reference. Each method generates $1500$ particles, and marginal and pairwise densities are estimated via kernel density estimation.
• Figure 3: The effect of different adaptive step thresholds $\Delta L$ and particle numbers $N$ on the validation dataset in terms of SWD and computational time.
• Figure 4: The likelihood landscape of the TDOA measurement. The nonlinear measurement equation creates a hyperbolic high-probability ridge.
• Figure 5: Qualitative comparison on a sample with an informative prior. The analytic flows struggle to capture the "banana" shape (Mean Exact) or properly cover the variance (Incompressible), while the Neural Flow matches the posterior geometry accurately.