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

This paper proposes a localization framework that uses spatiotemporal fields governed by partial differential equations (PDEs) as localization signatures. Two PDE classes are considered: the shallow water equations, which describe free-surface flows in coastal and riverine environments, and the advection-diffusion equation, which models the transport and mixing of scalar quantities such as temperature, salinity, and dissolved oxygen. A numerical PDE solver provides predicted fields over the domain, and multiple field channels are fused as multimodal measurements to improve localization accuracy. We formulate the problem within a Rao-Blackwellized particle filter (RBPF) that partitions the vehicle state into a nonlinear component sampled by particles and a linear sensor bias component tracked analytically via per-particle Kalman filters. This factorization reduces the required number of particles compared to a standard particle filter while accounting for realistic sensor drift. Simulation studies on both PDE scenarios show that the RBPF consistently outperforms a standard particle filter in terms of final position error and Root Mean Square Error (RMSE) across varying particle counts. Field experiments with an autonomous surface vehicle measuring salinity, temperature, and dissolved oxygen validate that PDE-governed environmental fields provide sufficient spatial variability for practical localization. Related experimental videos are available at https://localization-environmental-pdes.github.io/.

Sensor-rich data-driven applications increasingly use Bayesian approaches to infer latent states of dynamic systems from noisy sensor measurements and physical models. Yet the computation of the likelihood remains an essential bottleneck for accurate posteriors and performant inference. This paper presents a Bayesian filtering technique that uses processor-native uncertainty tracking for both uncertainty propagation and inference. The technique implements deterministic hierarchical importance restructuring through a native operation, giving deterministic latency and bounded memory use for arbitrary models written as program code. Benchmarks across three nonlinear state-space systems compare the approach against particle filters and Monte-Carlo-based likelihood estimators. The technique enables deterministic approximate filtering with as high as 805$\times$ average speedup against direct Monte Carlo work at matched result quality for model evaluation, and Pareto-dominant accuracy-latency trade-offs for posterior inference while remaining competitive in RMSE with baseline particle filters.

Deep gaze estimation works well in controlled capture but degrades in unconstrained settings, where systems must reject unreliable predictions. Single-pass uncertainty (e.g., heteroscedastic regression) infers uncertainty from pixels without explicit input-validity cues, while sampling based methods are often too costly for real time use. We propose Factor-Informed Uncertainty Distillation (FIUD), a teacher-student framework that aligns uncertainty with interpretable image-quality failure modes. A gradient-boosting teacher predicts expected gaze error from factors such as illumination, sharpness, eye visibility and symmetry; a neural student distills these signals via curriculum learning and ranking supervision into a lightweight single-pass uncertainty head. Across ETH-XGaze, Gaze360, and MPIIFaceGaze (>300k samples), FIUD improves uncertainty, error rank correlation and selective prediction versus deterministic and sampling-based baselines, with the largest gains in unconstrained settings.