A synergistic framework integrating CPO-VMD with BiLSTM-TimesNet for accurate prediction of nonlinear and nonstationary runoff time series.

Accurate runoff prediction is essential for effective water resource management, yet the complex characteristics of runoff sequences-such as nonlinearity, non-stationarity, and intricate temporal dependencies-pose significant challenges. This study proposes a hybrid prediction framework that integrates the Crested Porcupine Optimizer (CPO), Variational Mode Decomposition (VMD), Bidirectional Long Short-Term Memory (BiLSTM), and Temporal 2D-Variation Modeling for General Time Series Analysis (TimesNet). With the minimum envelope entropy as its objective, the CPO adaptively optimizes key VMD parameters to decompose raw sequences efficiently, generating refined input features for subsequent models. BiLSTM captures bidirectional temporal dependencies, and TimesNet performs multi-scale periodic feature analysis, thereby constituting a feature learning framework. The proposed hybrid model aggregates predictions from each sequence component to deliver accurate runoff forecasts. The study utilizes daily runoff data from the Quinebaug River in the United States and the Elbe River in Germany for validation (Quinebaug: 1997-2001; Elbe: 2019-2022). Comparative analysis shows that the proposed model outperformed the benchmark models across all evaluation metrics. Taking the Quinebaug station as an example, compared to the LSTM model, the Nash-Sutcliffe efficiency coefficient (NSE) improved by 16.15%, the Kling-Gupta efficiency coefficient (KGE) improved by 19.34%, the root mean square error (RMSE) decreased by 60.38%, and the mean absolute percentage error (MAPE) decreased by 72.56%. The results indicate that this model possesses significant advantages in forecasting nonlinear, non-stationary runoff series and can effectively improve prediction accuracy and stability. Although there is room for improvement in terms of model interpretability, the three-stage collaborative framework of "parameter optimization-signal decomposition-deep modeling" proposed in this study provides a reference technical solution for forecasting complex hydrological time series.