  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.
