# Physics-informed sensor coverage through structure preserving machine learning

arXiv: https://arxiv.org/abs/2509.10363 · HTML: https://mlsanity.com/paper/2509.10363/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 70 references that are in the map.
- First posted on arXiv 12 Sep 2025.

## Builds on (most cited first, 18 of 18)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Deep learning architectures for nonlinear operator functions and nonlinear inverse problems](https://mlsanity.com/paper/1912.11090/) (2019)
- [Structure-Preserving Digital Twins via Conditional Neural Whitney Forms](https://mlsanity.com/paper/2508.06981/) (2025)
- [On the definition and importance of interpretability in scientific machine learning](https://mlsanity.com/paper/2505.13510/) (2025)
- [Multi-Robot Control Using Time-Varying Density Functions](https://mlsanity.com/paper/1404.0338/) (2014)
- [Model-Based Active Source Identification in Complex Environments](https://mlsanity.com/paper/1706.01603/) (2017)

## Built on by (most cited first, 3 of 3)

- [Structure-Preserving Learning Improves Geometry Generalization in Neural PDEs](https://mlsanity.com/paper/2602.02788/) (2026)
- [Multi-robot Multi-source Localization in Complex Flows with Physics-Preserving Environment Models](https://mlsanity.com/paper/2509.14228/) (2025)
- [Neural Navigation Functions for Zero-Shot Generalizable Motion Planning](https://mlsanity.com/paper/2606.03756/) (2026)

## Abstract (arXiv)

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.