Probabilistic sensor placement in water distribution networks: a Dynamic Residual Masking framework using game-theoretic pseudo-Jacobians

Optimal sensor placement in Water Distribution Networks (WDNs) traditionally relies on rigid sensitivity matrices requiring computationally prohibitive white-box hydraulic simulations. While data-driven alternatives utilizing game-theoretic machine learning, specifically, Shapley Additive Explanations (SHAP), offer computational relief, they inherently suffer from a “dimensional trap.” Raw SHAP values mathematically inherit the target variable’s dimensions, creating a dimensional incompatibility when applied to network hydraulics. To resolve this, a fully data-driven, probabilistic sensor placement framework is proposed. Random Forest (RF) regressors are trained on stochastic, non-stationary network data to extract these game-theoretic marginal contributions. Crucially, a novel “pseudo-Jacobian” transformation is derived that restores the physical dimensional integrity of the SHAP-derived sensitivities. These mathematically valid sensitivities are then mapped through a Gaussian noise Cumulative Distribution Function (CDF) to create a strict estimated probability of detection landscape, abandoning arbitrary deterministic thresholds. Finally, a Dynamic Residual Masking algorithm is introduced. A soft-masking greedy search tracks unmet network estimated probabilities to naturally repel highly correlated sensor pairs, commonly referred to in optimization literature as the “evil twin” phenomenon. Validation on the publicly available 500-scenario Hanoi WDN benchmark demonstrates that the proposed framework bypasses the computational bottleneck of traditional finite-difference simulations. Furthermore, the methodology achieves a 91.60% accuracy in faulty/non-faulty scenario detection with highly sparse coverage (5 sensors), closely approaching the 93.80% accuracy yielded by full network coverage (31 sensors). Concurrently, a 2.5% reduction in false positives is achieved compared to the classical explicit-Jacobian baseline. This establishes a rigorous, mathematically valid bridge between game-theoretic artificial intelligence and physical hydraulic engineering.