Deep Probabilistic Indoor Gas Source Localization via Physical Dependency-Guided Sequential Inference

Abstract: Reliable gas source localization (GSL) is critical to safety in industrial and urban environments, yet remains challenging indoors because walls and obstacles interact with airflow to create complex gas dispersion. High-fidelity models such as computational fluid dynamics and filament models can capture these effects, but their computational cost limits online use. We propose a deep probabilistic framework that infers the source posterior from sparse and noisy measurements collected by a mobile robot. Unlike end-to-end models that directly infer source estimates from measurements, the proposed method incorporates physical dependencies of indoor gas transport, where wind and source location govern the concentration field. These dependencies are embedded through sequential conditional inference, in which inferred wind and concentration fields guide source posterior estimation. This structure improves localization under sparse and noisy observations. Evaluations show that the proposed method outperforms representative GSL baselines and enables accurate and efficient active GSL in simulations. Real-robot experiments demonstrate the feasibility of online operation on an embedded GPU.

• The paper introduces DGSE-S, a sequential U-Net framework that models wind, concentration, and source dependencies with probabilistic intermediate fields and Monte Carlo inference.
• The method achieves 100% active-search success across six simulated environments and controlled robot trials, with approximately 0.09–0.13-second inference compared with 7.3–8.7 seconds for PMFS.
• The results show sub-meter out-of-distribution localization accuracy and strong robustness, while revealing reduced path efficiency under domain shift and limitations for multi-source or transient dispersion.

Motivation and problem setting

Indoor gas source localization (GSL) requires an observation model that relates sparse, noisy gas and wind measurements to a hypothesized source location. High-fidelity dispersion models such as computational fluid dynamics (CFD) capture the complex transport induced by walls, obstacles, and forced airflow, but are too slow for online inference; filament-based models, while lighter, still require several seconds per posterior update because they must propagate stochastic filaments across many source hypotheses (2608.16221). End-to-end deep learning avoids repeated dispersion evaluations, but existing methods have been validated mainly in simplified, obstacle-free, or single-room settings and do not natively provide the probabilistic outputs needed for uncertainty-aware active search.

The paper addresses this gap with a deep probabilistic framework for single-source, steady-state GSL in known multi-room indoor layouts. The environment is discretized into $N$ cells at 0.1 m resolution; the source is a categorical variable over these cells, and a U-Net backbone maps rasterized wind measurements, log-transformed concentration measurements, and a binary occupancy grid to probabilistic outputs.

Architecture: from end-to-end to sequential conditional inference

Three variants are developed. DGSE-O directly predicts the source posterior $\mathrm{Cat}(s \mid \pi_\theta)$ from observations and map, serving as an end-to-end baseline comparable to prior work. DGSE-P uses a shared encoder with three decoders that estimate the wind field (WF), concentration field (CF), and source location (SL) in parallel as auxiliary tasks; physical dependencies are captured only implicitly through the shared latent representation. Field predictors output diagonal Gaussians, providing heteroscedastic aleatoric uncertainty per cell—a deliberate simplification that sacrifices spatial correlation structure for online tractability.

DGSE-S, the proposed method, encodes the causal dependency structure of indoor dispersion as a Bayesian network: the map and inlet velocity govern the wind field; the source location, release rate, map, and wind field govern the concentration field; measurements are generated from the fields. Two conditional independence relations follow—wind measurements add no information about $c$ given $(w, z^c, o)$, and sensor measurements add none about $s$ given $(w, c, o)$. These yield marginalizations:

$$p(s \mid z^w, z^c, o) = \mathbb{E}_{w^*}\left[\mathbb{E}_{c^* \sim p(c|w^*,z^c,o)}\left[p(s \mid w^*, c^*, o)\right]\right]$$

implemented by three U-Net subnetworks connected through reparameterized samples of the intermediate fields. At inference, expectations over the latent fields are approximated with $K=T=5$ Monte Carlo samples, GPU-batched for parallelism; the resulting Gaussian mixture for the concentration field is moment-matched to a single diagonal Gaussian.

Because the marginal likelihood contains logarithms of expectations, its finite-sample Monte Carlo estimate is biased. The authors instead optimize a Jensen's inequality lower bound, whose single-sample reparameterized estimator is unbiased. They report that increasing sample counts during training gave negligible improvement, and appendix analysis shows the surrogate's training trend matches Monte Carlo estimates of the original objective (losses stabilize once reference sample counts reach five). Tighter bounds such as IWAE were considered but deemed unnecessary.

Active GSL integration

The probabilistic outputs feed a deliberately simple utility-based planner rather than a novel search algorithm:

$$U_i = \exp(-w_d \cdot dist_i)\,(w_w\sigma_{w,i}^2 + w_c\sigma_{c,i}^2 + w_s\pi_{s,i})$$

with fixed weights $(0.1, 0.25, 2.0, 100.0)$ across all environments, Dijkstra path distances, and entropy-based termination at $\mathrm{Cat}(s \mid \pi_\theta)$0. The stated purpose is to test whether the estimated posteriors and field uncertainties carry enough information to guide search—not to optimize the policy itself.

Training data

Training uses 3,750 procedurally generated multi-room layouts (5–10 m per side), simulated in OpenFOAM with RANS $\mathrm{Cat}(s \mid \pi_\theta)$1–$\mathrm{Cat}(s \mid \pi_\theta)$2 closure for wind and passive-scalar advection–diffusion for ethanol dispersion, yielding 15,000 environments split at the layout level into 12,000/2,000/1,000 training/validation/test. Inlet velocities span 0.5–5.0 m/s and release rates 0.01–5.0 g/s. Sensor noise is randomized per sample: multiplicative-plus-environmental Gaussian noise on concentrations ($\mathrm{Cat}(s \mid \pi_\theta)$3), multiplicative speed noise up to 50%, and direction noise up to 15°, covering ranges used in prior GSL studies and commercial anemometer specifications. Each epoch draws 30 fresh random sensor configurations per environment (5–300 sensors), producing 360,000 distinct pairs per epoch, augmented by six symmetry transforms. Notably, sensors are placed randomly rather than along trajectories—an intentional choice to avoid overfitting to trajectory shapes, which the authors show transfers successfully to sequential active-search observations.

Results

Variant comparison. Across 1,000 test environments with varying observation counts, DGSE-P improves over DGSE-O, indicating that intermediate-field supervision stabilizes source learning under sparse noisy data (DGSE-O alone exhibits unstable validation loss despite decreasing training loss). DGSE-S further outperforms DGSE-P on both CF prediction (RMSE and NLL) and SL (localization error and cross-entropy), supporting the claim that explicitly encoding the physical dependency structure—rather than relying on implicit shared-representation coupling—improves inference.

GDM comparison. Against KDM+V/W, GaBP+, G-GMRF, and GW-GMRF on five test environments at 50–250 observations, DGSE-S achieves the lowest RMSE among all methods and the lowest NLL among methods providing comparable probabilistic outputs. Predicted standard deviation correlates spatially with actual reconstruction error. On deployment-appropriate hardware (Jetson AGX Orin for DGSE-S versus an Intel NUC CPU for classical baselines), DGSE-S has the lowest latency.

Static source localization. Using only estimation components (no planning) with 50 random sensing locations over six GADEN environments, DGSE-S attains lower localization error than PMFS (filament-based) and the dual-mode planner (particle filter with a low-airflow analytical model) in all six environments.

Active GSL simulation. Over 20 trials per environment, DGSE-S achieves a 100% success rate in all six GADEN environments, and among methods tied on success rate it records the lowest error, shortest search time, shortest travel distance, and fastest one-step computation (~0.09–0.13 s versus ~7.3–8.7 s for PMFS). GrGSL fails entirely in Environment 4 (0% SR), attributed to premature convergence driven by unreliable local upwind information. An out-of-domain evaluation in six environments with unseen layouts, obstacles, dimensions, and inlet/outlet configurations retains 100% success rate and sub-meter mean error, though the mean path efficiency ratio degrades from 1.89 in-domain to 3.57 out-of-domain—successful but less efficient search outside the training distribution.

Real-robot experiments. On an Agilex LIMO UGV with FAST-LIO SLAM, an NDIR CO$\mathrm{Cat}(s \mid \pi_\theta)$4 sensor, and a TriSonica LI-550 anemometer, CO$\mathrm{Cat}(s \mid \pi_\theta)$5 substitutes for ethanol (justified by similar molecular weight and diffusivity, with a fixed input rescaling). In two contrasting scenarios (downstream-to-upstream and upstream-to-downstream starts, 10 trials each), DGSE-S again achieves 100% success rate, with lower error, shorter search time, and shorter travel distance than both baselines. The paper attributes PMFS's total failure in Experiment 1 (0% SR) to its binary hit-map thresholding, which cannot distinguish the primary plume from secondary accumulation regions, whereas DGSE-S reasons over continuous concentration fields. The dual-mode planner suffers from model mismatch, converging toward high-concentration transport regions rather than the source under airflow-driven dispersion. The authors are explicit that these experiments constitute controlled validation under structured fan-driven airflow, not exhaustive tests across arbitrary real-world dispersion regimes.

Limitations and open questions

Several constraints bound the reported results. The framework assumes single-source, steady-state dispersion with a known map and a single inlet/outlet flow regime matching the training generator; multi-source and transient scenarios are deferred. The diagonal-Gaussian field parameterization discards spatial correlations between cells, which may matter for downstream uncertainty quantification. The Jensen lower bound is validated empirically against Monte Carlo estimates only within the evaluated setting, not proven tight. Real-world validation covers one laboratory configuration with a fan-established inlet and a CO$\mathrm{Cat}(s \mid \pi_\theta)$6/ethanol substitution mediated by a fixed scale factor. Finally, the OOD results show that generalization preserves accuracy but not search efficiency, leaving open how planners can be made robust to domain shift without sacrificing trajectory efficiency.

Conclusion

This work presents a deep probabilistic GSL framework whose central design decision—decomposing source inference into sequential conditioning on inferred wind and concentration fields, derived from the physical dependency structure of indoor dispersion—yields consistent empirical gains over both parallel multi-task and end-to-end alternatives. Combined with CFD-scale randomized training data and lightweight utility-based planning, it delivers 100% success rates in simulated and controlled real-world active GSL with sub-second embedded-GPU inference cycles, addressing the principal computational obstacle that has limited high-fidelity-model-based indoor GSL to offline use.