Physics-Informed Topological Signal Processing for Water Distribution Network Monitoring

Abstract: Water management is one of the most critical aspects of our society, together with population increase and climate change. Water scarcity requires a better characterization and monitoring of Water Distribution Networks (WDNs). This paper presents a novel framework for monitoring Water Distribution Networks (WDNs) by integrating physics-informed modeling of the nonlinear interactions between pressure and flow data with Topological Signal Processing (TSP) techniques. We represent pressure and flow data as signals defined over a second-order cell complex, enabling accurate estimation of water pressures and flows throughout the entire network from sparse sensor measurements. By formalizing hydraulic conservation laws through the TSP framework, we provide a comprehensive representation of nodal pressures and edge flows that incorporate higher-order interactions captured through the formalism of cell complexes. This provides a principled way to decompose the water flows in WDNs in three orthogonal signal components (irrotational, solenoidal and harmonic). The spectral representations of these components inherently reflect the conservation laws governing the water pressures and flows. Sparse representation in the spectral domain enable topology-based sampling and reconstruction of nodal pressures and water flows from sparse measurements. Our results demonstrate that employing cell complex-based signal representations enhances the accuracy of edge signal reconstruction, due to proper modeling of both conservative and non-conservative flows along the polygonal cells.