Enhancing process-based hydrological models with embedded neural networks: A hybrid approach

Deep learning (DL) models have demonstrated exceptional performance in hydrological modeling; however, they are limited by their inability to output untrained hydrological variables and lack of interpretability compared to process-based hydrological models. We propose a hybrid approach that combines the conceptual EXP-Hydro model with embedded neural networks (ENNs), replacing its internal modules while maintaining adherence to hydrological knowledge. The resulting hybrid model can predict untrained hydrological variables without requiring post-processing or pre-training procedures. We tested 15 hybrid models that replace different internal modules across 569 basins in the contiguous United States using the CAMELS dataset. Additional experiments were conducted to generalize hydrological relationships within ENNs and further use them to improve the EXPHydro model's performance. Results show that all hybrid scenarios outperform the ordinary EXP-Hydro model, with an optimal median Nash-Sutcliffe efficiency (NSE) of 0.701 in the evaluation period - comparable to stateof-the-art LSTM and conceptual hydrological model featuring an error-correcting post-processor. Reasonable patterns of runoff and snow-related processes are captured by ENNs in respective hybrid models. We further used the runoff (snow-related) pattern to improve the ordinary EXP-Hydro model with median NSE increasing from 0.496 to 0.567 (raising median NSE from 0.601 to 0.677 in snow-influenced region). Our study highlights the potential for using ENNs in enhancing process-based hydrological models' performance while maintaining interpretability within a novel hybrid framework.