Hybrid-GR4J: A hybrid hydrological model integrating GR4J and deep learning

Deep learning models have shown outstanding performance in hydrological modeling but are often questioned for their “black-box” nature. To address this issue, this study proposes Hybrid-GR4J, a hybrid model that embeds the structure of the GR4J hydrological model into a physics-constrained recurrent neural network, enabling end-to-end training within the NODE framework. The model is evaluated using daily meteorological inputs across 569 catchments from the CAMELS-US dataset. Results indicate that Hybrid-GR4J achieves average NSE and KGE scores of 0.59 and 0.63, representing improvements of 23.52 % over RNN and 36.58 % over GR4J, respectively. Moreover, the model exhibits strong robustness under various climatic conditions and training data lengths. This study confirms the effectiveness of structurally embedded hybrid modeling in improving runoff simulation accuracy and provides a transferable framework for integrating physical knowledge with data-driven approaches.