Hydrological modeling with priors information and deep learning under small-sample conditions

Hydrological modeling is often constrained by small-sample problems in certain regions, leading to a significant decline in performance. To address this issue, this paper proposes a Gamma Convolutional Neural Network (GCN) that integrates hydrological prior information. It characterizes the time-lag response mechanism of rainfall—runoff using the Gamma distribution probability density function. A runoff weight allocation structure is incorporated to explicitly represent basin storage effects and multi-timescale runoff components. Moreover, GCN jointly optimizes hydrophysical parameters and network weights in an end-to-end manner to achieve adaptive parameter learning. In the test conducted at the Shijiao Station in the Beijiang River Basin, the multi-kernel GCN network achieved a Nash-Sutcliffe efficiency coefficient (E_NS) of over 0.90 under full-sample conditions, outperforming the Xin’anjiang Model (XAJ) and Multilayer Perceptron (MLP). For a minimal sample size of 5%, E_NS maintained a value of 0.86 (XAJ: 0.67, MLP: 0.73). The obtained Gamma parameter exhibited regional adaptability, offering a plausible interpretation of the runoff’s time-lag characteristics. The study demonstrates that GCN effectively improves both data efficiency and interpretability in small-sample hydrological modeling.