Distill-Belief: Closed-Loop Inverse Source Localization and Characterization in Physical Fields

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 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.