Reservoir networks must be monitored continuously to avert costly water losses and service disruptions, yet existing anomaly-detection models often ignore the governing physics and thus yield hydrologically infeasible alarms. We introduce Temporal Graph Attention Network with Physics-Informed Constraints (TGAT-PIC)—a real-time anomaly detector that unifies graph neural learning with explicit hydrological knowledge. TGAT-PIC (i) builds a hierarchical graph that fuses physical connectivity, operational dependencies, and learned correlations; (ii) couples graph attention with dilated causal convolutions to capture multi-scale spatial–temporal patterns; and (iii) embeds three complementary physics constraints into both training and inference: water-balance conservation, capacity-feasibility projection, and soft rule-curve regularization. Extensive experiments on the 679-reservoir ResOpsUS benchmark and three live California reservoir systems show that TGAT-PIC achieves 96.8% F1-score, surpassing the best state-of-the-art baseline by 8.3% while cutting detection delay to 1.3 h. Field deployments reduce verified water losses by 23% and raise operational efficiency by 31%, directly advancing UN SDG 6 targets for water-use efficiency and service reliability. Code, data splits, and trained models will be released upon publication to facilitate reproducibility and further research.