Estimating associations between spatial covariates and responses -- rather than merely predicting responses -- is central to environmental science, epidemiology, and economics. For instance, public health officials might be interested in whether air pollution has a strictly positive association with a health outcome, and the magnitude of any effect. Standard machine learning methods often provide accurate predictions but offer limited insight into covariate-response relationships. And we show that existing methods for constructing confidence (or credible) intervals for associations can fail to provide nominal coverage in the face of model misspecification and nonrandom locations -- despite both being essentially always present in spatial problems. We introduce a method that constructs valid frequentist confidence intervals for associations in spatial settings. Our method requires minimal assumptions beyond a form of spatial smoothness and a homoskedastic Gaussian error assumption. In particular, we do not require model correctness or covariate overlap between training and target locations. Our approach is the first to guarantee nominal coverage in this setting and outperforms existing techniques in both real and simulated experiments. Our confidence intervals are valid in finite samples when the noise of the Gaussian error is known, and we provide an asymptotically consistent estimation procedure for this noise variance when it is unknown.

In this supplementary material we provide additional details about: Video (with audio) for qualitative illustration of our task and qualitative evaluation of our model predictions (Sec. Evaluation of the impact of the query source location on our model's prediction quality for a fixed receiver (Sec. Moreover, we qualitatively demonstrate our model's prediction quality by comparing the predictions with the ground truths, both at the RIR level and in terms of perceptual similarity when the RIRs are convolved with real-world monaural sounds, like speech and music. We also analyze common failure cases for our model (Sec. Please use headphones to hear the spatial audio correctly.

Usingcollectedindirect measurements and the forward model, the goal becomes to recover the source. When the forward model is unknown, or imperfect, artifacts due to model mismatch occur in the recovery of the source.

High-resolution imaging from array data in unknown inhomogeneous ambient media requires estimating both the medium properties and the object characteristics. For diverse measurements collected from different sources in different, changing media, we introduce in this paper an algorithm that recovers the ambient media properties needed for high-resolution imaging as well as the source locations and strengths that constitute the imaging target. This algorithm extends and improves upon our previous work on imaging through random media using array data. Previously, we addressed imaging through a single unknown random medium, either weakly scattering [ 1 ] or strongly scattering [ 2 ].

Abstract-- This paper proposes an uncertainty-aware marine pollution source tracking framework for unmanned surface vehicles (USVs). By integrating high-fidelity marine pollution dispersion simulation with informative path planning techniques, we demonstrate effective identification of pollution sources in marine environments. The proposed approach is implemented based on Robot Operating System (ROS), processing real-time sensor data to update probabilistic source location estimates. Experiments conducted in simulated environments with varying source locations, wave conditions, and starting positions demonstrate the framework's ability to localise pollution sources with high accuracy. Results show that the proposed approach achieves reliable source localisation efficiently and outperforms the existing baseline. This work contributes to the development of full autonomous environmental monitoring capabilities essential for rapid response to marine pollution incidents. Pollution discharged into the marine environment causes severe consequences to ecosystems [1], [2] and human health [3].

We present a machine learning framework for adaptive source localization in which agents use a structure-preserving digital twin of a coupled hydrodynamic-transport system for real-time trajectory planning and data assimilation. The twin is constructed with conditional neural Whitney forms (CNWF), coupling the numerical guarantees of finite element exterior calculus (FEEC) with transformer-based operator learning. The resulting model preserves discrete conservation, and adapts in real time to streaming sensor data. It employs a conditional attention mechanism to identify: a reduced Whitney-form basis; reduced integral balance equations; and a source field, each compatible with given sensor measurements. The induced reduced-order environmental model retains the stability and consistency of standard finite-element simulation, yielding a physically realizable, regular mapping from sensor data to the source field. We propose a staggered scheme that alternates between evaluating the digital twin and applying Lloyd's algorithm to guide sensor placement, with analysis providing conditions for monotone improvement of a coverage functional. Using the predicted source field as an importance function within an optimal-recovery scheme, we demonstrate recovery of point sources under continuity assumptions, highlighting the role of regularity as a sufficient condition for localization. Experimental comparisons with physics-agnostic transformer architectures show improved accuracy in complex geometries when physical constraints are enforced, indicating that structure preservation provides an effective inductive bias for source identification.

To allow for scalable inference, we implement our model as a variational autoencoder and perform amortized variational inference.