A Coupled Model Integrating a Differential Equation and BP Neural Network for Seepage Pressure Forecasting of Earth-Rock Dams

Reliable daily seepage pressure forecasting remains challenging for earth-rock dam safe ty monitoring. This study aimed to develop a hybrid framework coupling back-propagation neural network (BPNN) and differential equation (DE) for daily seepage pressure prediction. The framework was validated using daily monitoring records (January 2020–July 2024) from three piezometers at the Wangqingtuo regulating reservoir, Tianjin. Input variables included reservoir water level, squared water level, precipitation, and maximum and minimum air temperature. The dataset was partitioned into training (2020–2022), internal testing (2022–2023), and external forecasting (2023–2024) subsets. The two-stage method first trains a BPNN to produce baseline predictions and extract residuals; then particle swarm optimization (PSO) calibrates unknown coefficients of the residual-based DE. Among DE variants, BPNN–DE-1TD achieves the most stable performance across monitoring points. Compared with standalone BPNN, this model reduces mean absolute percentage error (MAPE) by 17%, 19%, and 26% at P5-1, P4-2, and P6-4, yielding absolute MAPE of 1.56%, 1.17%, and 5.41%, respectively. The hybrid framework effectively captures seasonal oscillations and time-lag responses of seepage pressure. This work demonstrates the value of residual-corrected semi-physical modelling for controlled shallow regulating reservoirs, and the method may be adapted for comparable dam seepage forecasting cases with similar hydraulic characteristics.