Stacked hybridization of deep learning model with grey wolf optimization for accurate and explainable reference evapotranspiration

Accurate estimation of reference evapotranspiration (ET₀) is essential for effective irrigation scheduling and water resource management, particularly in data-scarce regions such as India, which lack advanced automatic meteorological stations. The present study developed a hybrid model (DNN-GWO) and conducted an in-depth evaluation against standalone data-driven models, including Random Forest (RF), Support Vector Machine (SVM), Deep Neural Network (DNN), Recurrent Neural Network (RNN), and Deep Belief Network (DBN) for forecasting monthly ET₀ in the Upper Ganga canal command region, Uttar Pradesh, India. Three input scenarios were evaluated for their correlation to ET₀ estimation. The results revealed that the DNN model showed the best performance in all three scenarios, achieving R² = 0.958, RMSE = 0.076 mm/day, NSE = 0.954, RMSLE = 0.024, MAE = 0.055, MBE = 0.012, MSRE = 0.032, and EVS = 0.987 with solar radiation (Rs), wind speed (U), maximum temperature (Tmax), minimum temperature (Tmin), and relative humidity (RH) as inputs. The developed hybrid DNN-GWO model further improved predictive accuracy, with R² = 0.992, RMSE = 0.0317 mm/day, NSE = 0.99, RMSLE = 0.023, MAE = 0.054, MBE = 0.018, and EVS = 0.992, reducing RMSE by nearly 60 % compared to the best-performing standalone DNN. SHapley Additive explanations (SHAP) analysis revealed that temperature and solar radiation were the most influential predictors of ET₀, while the model also provided stable predictions across different input scenarios, demonstrating robustness in data-limited conditions. The developed hybrid framework, by combining deep learning, swarm intelligence, and explainability, provides a robust, accurate, and interpretable solution for agricultural water management in data-constrained environments.