Physics-informed neural networks in water and wastewater systems: a critical review.

Physics-Informed Neural Networks (PINNs) represent a hybrid modeling paradigm that embeds governing physical laws, expressed as partial differential equations (PDEs), directly into neural network training. This integration enables models to respect fundamental conservation principles while learning from sparse or incomplete data. This review critically examines PINN applications in water and wastewater systems over the period 2014-2024, focusing on drinking water distribution networks, wastewater treatment plants, urban drainage systems, and water treatment processes. The review shows that PINNs excel in inverse problem solving by enabling parameter estimation and system identification from indirect observations, while maintaining physical consistency in extrapolation regimes where purely data-driven models fail. Documented applications report performance advantages, including 3-30 × reductions in required training data compared to standard neural networks, improved generalization under distribution shift, and successful use in scenarios involving partial observations and uncertain boundary conditions. However, critical limitations emerge: PINNs require well-posed problems with reliable governing equations, struggle with complex networked systems involving discrete components, face major convergence challenges for stiff or multi-scale PDEs, and still lack mature uncertainty quantification frameworks. Rather than positioning PINNs as replacements for established numerical methods, this work frames them as complementary tools that bridge mechanistic modeling and data-driven learning, offering particular value in parameter calibration, sensor placement optimization, and real-time state estimation for water infrastructure systems.