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arXiv · 2509.10363

Physics-informed sensor coverage through structure preserving machine learning

Benjamin David Shaffer·Brooks Kinch·Joseph Klobusicky·M. Ani Hsieh·Nathaniel Trask
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Abstract

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.

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Benjamin David Shaffer, Brooks Kinch, Joseph Klobusicky, M. Ani Hsieh, Nathaniel Trask. 2025-09-12. Physics-informed sensor coverage through structure preserving machine learning. https://arxiv.org/abs/2509.10363

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