Population-Aware Physics-Informed Neural Particle Flow for Robust Spacecraft Bayesian Navigation

Spacecraft navigation often requires Bayesian inference from sparse nonlinear measurements that produce curved, multimodal, or geometrically constrained posterior distributions. Physics-informed neural particle flow (PINPF) addresses such problems by learning a deterministic prior-to-posterior transport field from the governing probability evolution equation, but its particle-wise architecture does not explicitly account for the empirical particle population. This paper introduces population-aware PINPF (PA-PINPF), which conditions each particle velocity on a permutation-invariant Deep Sets representation of the complete particle set. PA-PINPF-State summarizes particle positions, whereas PA-PINPF-Feature summarizes the local physics-informed features, including likelihood and score information. Both retain the unsupervised PINPF residual and require neither posterior samples nor labeled transport trajectories. The methods are evaluated over 100 randomized time-difference-of-arrival tasks representative of RF cross-link and ground-station localization and 100 range-measurement tasks with an onboard sensor field-of-view constraint. Results demonstrate that population-level Bayesian features provide useful global information for learned particle transport in nonlinear spacecraft-navigation problems.