Localization in Spatiotemporal Fields via Environmental PDEs

Maritime target tracking over large distances often requires multi-agent teams without centralized coordination, and intermittent communication. Each agent must maintain an independent estimate that can take advantage of opportunistic communications availability when possible. This can lead to overly confident local estimates in the absence of external data. In this work, we propose an augmentation to a classical particle filter implementation that accounts for this potential source of error by forcing particles to spread strategically in the absence of informative updates from other sensor nodes. We demonstrate our method using Unmanned Surface Vessels (USVs) on a lake, and show that our augmentations do not deteriorate nominal performance, and provide an advantage in some specific edge cases.

{Closed-loop inverse source localization and characterization (ISLC) requires a mobile agent to select measurements that localize sources and infer latent field parameters under strict time constraints.} {The core challenge lies in the belief-space objective: valid uncertainty estimation requires expensive Bayesian inference, whereas using fast learned belief model leads to reward hacking, in which the policy exploits approximation errors rather than actually reducing uncertainty.} {We propose \textbf{Distill-Belief}, a teacher--student framework that decouples correctness from efficiency. A Bayes-correct particle-filter teacher maintains the posterior and supplies a dense information-gain signal, while a compact student distills the posterior into belief statistics for control and an uncertainty certificate for stopping. At deployment, only the student is used, yielding constant per-step cost.} {Experiments on seven field modalities and two stress tests show that Distill-Belief consistently reduces sensing cost and improves success, posterior contraction, and estimation accuracy over baselines, while mitigating reward hacking.}

Accurate and robust localization is essential for autonomous mobility systems in real-world environments. While fusing Inertial Measurement Unit (IMU) data with satellite-based correction signals provides precise vehicle pose estimates, performance degrades substantially during outages. Recent studies indicate that Machine Learning (ML) can improve IMU-based proprioceptive localization, highlighting untapped potential for onboard sensors readily available in production vehicles. This paper introduces Physics-Regularized Machine Learning for Localization (PRML2), a hybrid framework that combines the complementary strengths of Kalman filtering and data-driven learning to estimate vehicle pose directly from onboard sensors. A key aspect of PRML2 is its physics-regularized learning, enabled by end-to-end training of an ML model through a differentiable Kalman filter. This improves consistency with vehicle motion models, thereby enhancing both localization accuracy and generalization across driving conditions. We evaluate the performance limits of ML-enhanced onboard odometry on a publicly available dataset and show that PRML2 achieves superior localization accuracy and demonstrates real-time capability. This work also introduces a novel dataset to support vehicle localization research under low-friction conditions. The proposed framework provides a robust and cost-effective solution for vehicle localization under degraded sensing conditions by integrating learning with physics-based priors.