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. Our code and data are available under https://github.com/EPFL-IMOS/PEGNN_SS.