The Subtle Art of Problem Formulation: Fusing Physics into Graph Structures for GNN-Based Pressure Monitoring in Water Distribution Networks

Water Distribution Networks (WDNs) are vital for urban sustainability, but monitoring them is challenging due to sparse sensors and complex hydraulics. While traditional models handle forward problems well, reconstructing network states from limited measurements (the inverse problem) is less developed. Current machine learning methods often overlook physical laws. We introduce a novel framework that integrates hydraulic principles directly into the graph structure of a Graph Neural Network (GNN), enabling accurate reconstruction of nodal pressures from sparse data. Our approach includes a Dirichlet-based method for generating realistic demand and leakage scenarios and a new benchmarking protocol tailored to operational needs. Experiments on standard benchmarks demonstrate superior performance with fewer sensors compared to conventional GNNs, and our evaluation framework links model errors to network characteristics, providing actionable insights for deployment. This physics-infused GNN paradigm advances WDN monitoring and leak detection.