Physics-guided networks for probabilistic hydrodynamic forecasting in canal systems

Reliable prediction of water supply dynamics in large-scale canal systems is critical for water allocation and operational decision-making in inter-basin water transfer projects. Uncertainty in lateral offtake discharges evolves over time and often exhibits multi-peaked distributions due to real-time hydraulic states and unplanned gate operations. However, reliably quantifying and interpreting the evolving uncertainty remains difficult under such dynamically changing and small-sample conditions. Here we show that a physics-guided mixture density network (PgMDN) can effectively characterize this uncertainty while remaining physically consistent. In the proposed PgMDN, physical knowledge is incorporated into the loss function through local mass balance and a consistency constraint between predictions and their associated uncertainty, while long short-term memory layers are employed to model temporal dependencies and multi-factor influences. In addition, Shapley additive explanation analysis is used to identify the dominant hydraulic inputs contributing to predictive uncertainty. Tested on real-world canal datasets, the proposed PgMDN outperforms the standard mixture density network, achieving over a 25% reduction in both mean absolute error and root mean square error, together with improved reliability, as measured by the R-index (increasing from 0.45 to 0.82), and stronger generalization. The results further reveal that water level fluctuations and boundary inflow are key drivers of predictive uncertainty, supporting the physical interpretability of the proposed model. Overall, this study provides a scalable and interpretable tool for real-time modeling of environmental infrastructure and the operational management of large-scale water diversion systems.