A Mass-Conserving Spatiotemporal Graph ODE Model for Multi-Step Streamflow Forecasting in a Sparsely Gauged Semi-Arid Basin

Semi-arid loess basins in the middle Yellow River have rapid runoff responses, short hydrological memory, and sparse gauging networks, so pooled metrics across all stations can conceal forecast failures in low-flow tributaries and during dry periods. This study proposes DaSTGODE, a data-augmented spatiotemporal graph ordinary differential equation model with a structural water balance constraint. DaSTGODE integrates directed advection–diffusion propagation, soil-moisture-limited evapotranspiration, and a recursive water balance within a unified framework, thereby closing the water budget at every forecast step through the model structure. A reference scenario with perfect meteorological forcing, DaSTGODE_pre, is also established. It supplies the decoder with meteorological conditions over the forecast window as future meteorological forcing and quantifies the potential contributions of future precipitation, air temperature, and potential evapotranspiration by channel. Using daily records from 10 stations in the Wuding River basin from 2006 to 2020, DaSTGODE is compared with long short-term memory (LSTM) and graph long short-term memory (GraphLSTM) models for 1-, 3-, 6-, and 12-day forecast windows. DaSTGODE increases pooled Nash–Sutcliffe efficiency (NSE) from 0.532 and 0.473 for the best baseline to 0.599 and 0.567 for the 6- and 12-day windows, respectively. Relative to the best baseline, DaSTGODE increases the 1-day station-macro logarithmic NSE (logNSE) from 0.266 to 0.612 and reduces the mean number of stations with a negative NSE in the 6-day window from 4.6 to 2.0. Under the reference scenario in which future meteorological conditions are known, DaSTGODE_pre increases the 6-day outlet NSE from 0.141 for DaSTGODE to 0.331, and future precipitation alone provides 94% of the gain obtained from all three meteorological channels. HBV virtual nodes yield only small improvements in metrics at gauged stations, but provide daily streamflow forecasts at the outlets of 16 ungauged subbasins. Taken together, these results show that, under the semi-arid and sparsely gauged conditions examined here, the complete physical graph ODE kernel mainly improves cross-station reliability and relative low-flow dynamics, while the perfect meteorological forcing experiment quantifies the contribution of meteorological information over the forecast period to accuracy at medium and long forecast windows and identifies the channels responsible for that contribution.