A differentiable, physics-based hydrological model and its evaluation for data-limited basins

Abstract

Recent advancements in deep learning (DL) have significantly improved hydrological modeling by extracting generalities from large-sample datasets and enhancing predictive accuracy. However, DL models often rely heavily on large volumes of data, which are often unavailable or insufficient in many real-world hydrological applications. This challenge has prompted interest in integrating DL with physically based hydrological models (PBHMs). This study explores such integration using differentiable programming with the Xin’anjiang model. We introduce two advanced model variants: the differentiable Xin’anjiang model (dXAJ), which retains the Xin’anjiang model’s structure while incorporating Long Short-Term Memory (LSTM) networks for parameter learning, and the dXAJnn model, which replaces the traditional evapotranspiration module of dXAJ model with a neural network. Both models were evaluated against the evolutionary algorithm-calibrated XAJ model (eXAJ) across five basins in the Three Gorge region of China and eight basins from the CAMELS dataset under varying data-limited conditions. Our results showed that both dXAJ and dXAJnn models outperformed the eXAJ model in streamflow prediction accuracy as they have different optimization mechanism, demonstrating that the local optimization mechanism in differentiable models (DMs) tends to generalize better during validation than global optimization approaches in data-limited contexts. The DMs also provided reliable evapotranspiration estimates, even without using evapotranspiration data for calibration. Although the dXAJnn model offered greater flexibility, it did not consistently yield better results and exhibited a tendency toward overfitting in certain basins. The study also found that both models require a minimum of three years of training data (including a one-year warm-up period) to achieve acceptable predictive performance, with longer data records further preventing overfitting. These findings underscore the ability of DMs to effectively balance data-driven techniques and physical mechanisms, highlighting the importance of sufficient training data.