# Physics-Enhanced Graph Neural Networks For Soft Sensing in Industrial Internet of Things

arXiv: https://arxiv.org/abs/2404.08061 · HTML: https://mlsanity.com/paper/2404.08061/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 57 references that are in the map.
- First posted on arXiv 11 Apr 2024.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Masked Label Prediction: Unified Message Passing Model for Semi-Supervised Classification](https://mlsanity.com/paper/2009.03509/) (2020)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Graph Neural Network-Based Anomaly Detection in Multivariate Time Series](https://mlsanity.com/paper/2106.06947/) (2021)
- [BRITS: Bidirectional Recurrent Imputation for Time Series](https://mlsanity.com/paper/1805.10572/) (2018)
- [Data Augmentation for Graph Neural Networks](https://mlsanity.com/paper/2006.06830/) (2020)
- [A Survey on Graph Neural Networks for Time Series: Forecasting, Classification, Imputation, and Anomaly Detection](https://mlsanity.com/paper/2307.03759/) (2023)
- [FourierGNN: Rethinking Multivariate Time Series Forecasting from a Pure Graph Perspective](https://mlsanity.com/paper/2311.06190/) (2023)
- [Filling the G_ap_s: Multivariate Time Series Imputation by Graph Neural Networks](https://mlsanity.com/paper/2108.00298/) (2021)
- [Local Augmentation for Graph Neural Networks](https://mlsanity.com/paper/2109.03856/) (2021)
- [DyEdgeGAT: Dynamic Edge via Graph Attention for Early Fault Detection in IIoT Systems](https://mlsanity.com/paper/2307.03761/) (2023)
- [Graph-based Virtual Sensing from Sparse and Partial Multivariate Observations](https://mlsanity.com/paper/2402.12598/) (2024)
- [Virtual Sensor for Real-Time Bearing Load Prediction Using Heterogeneous Temporal Graph Neural Networks](https://mlsanity.com/paper/2404.02304/) (2024)

## Built on by (most cited first, 6 of 6)

- [Virtual Sensing-Enabled Digital Twin Framework for Real-Time Monitoring of Nuclear Systems Leveraging Deep Neural Operators](https://mlsanity.com/paper/2410.13762/) (2024)
- [Graph Neural Networks for Virtual Sensing in Complex Systems: Addressing Heterogeneous Temporal Dynamics](https://mlsanity.com/paper/2407.18691/) (2024)
- [Efficient Unsupervised Domain Adaptation Regression for Spatial-Temporal Sensor Fusion](https://mlsanity.com/paper/2411.06917/) (2024)
- [Time-Vertex Machine Learning for Optimal Sensor Placement in Temporal Graph Signals: Applications in Structural Health Monitoring](https://mlsanity.com/paper/2512.19309/) (2025)
- [RINS-T: Robust Implicit Neural Solvers for Time Series Linear Inverse Problems](https://mlsanity.com/paper/2510.17396/) (2025)
- [Virtual Smart Metering in District Heating Networks via Heterogeneous Spatial-Temporal Graph Neural Networks](https://mlsanity.com/paper/2604.10166/) (2026)

## Abstract (arXiv)

The Industrial Internet of Things (IIoT) is reshaping manufacturing, industrial processes, and infrastructure management. By fostering new levels of automation, efficiency, and predictive maintenance, IIoT is transforming traditional industries into intelligent, seamlessly interconnected ecosystems. However, achieving highly reliable IIoT can be hindered by factors such as the cost of installing large numbers of sensors, limitations in retrofitting existing systems with sensors, or harsh environmental conditions that may make sensor installation impractical. Soft (virtual) sensing leverages mathematical models to estimate variables from physical sensor data, offering a solution to these challenges. Data-driven and physics-based modeling are the two main methodologies widely used for soft sensing. The choice between these strategies depends on the complexity of the underlying system, with the data-driven approach often being preferred when the physics-based inference models are intricate and present challenges for state estimation. However, conventional deep learning models are typically hindered by their inability to explicitly represent the complex interactions among various sensors. To address this limitation, we adopt Graph Neural Networks (GNNs), renowned for their ability to effectively capture the complex relationships between sensor measurements. In this research, we propose physics-enhanced GNNs, which integrate principles of physics into graph-based methodologies. This is achieved by augmenting additional nodes in the input graph derived from the underlying characteristics of the physical processes. Our evaluation of the proposed methodology on the case study of district heating networks reveals significant improvements over purely data-driven GNNs, even in the presence of noise and parameter inaccuracies.