Mitigating underestimation risks and enhancing water level forecasting accuracy with a risk-weighted hybrid loss function.

Water level forecasting plays a critical role in flood early warning and water resource management. Although deep learning methods have been widely used in hydrological time series prediction, they still suffer from insufficient fitting for extreme events and systematic underestimation of high water levels, which weakens the reliability of risk warning systems. To address these issues, this study designs a risk-motivated hybrid loss function. It introduces underestimation and extreme penalty terms that statistically constrain risky forecast biases into mean squared error (MSE) or mean absolute error (MAE) baselines, with penalty coefficients adaptively optimized via Bayesian optimization. Using a bidirectional long short-term memory (BiLSTM) network as the predictor, an ablation study is conducted with four experimental groups: baseline (no penalty), underestimation penalty only, extreme penalty only, and combined penalty. Results show that: (1) The underestimation penalty term effectively corrects systematic bias through an asymmetric gradient mechanism; (2) The extreme penalty term improves the model's fitting capability and stability for extreme water levels through sample reweighting; (3) The combined penalty achieves synergistic optimization, delivering the greatest improvement under the MSE baseline, with overall RMSE reduced by 9.8% and extreme interval RMSE reduced by 18.1% compared to the baseline. It maintains a relative low underestimation rate while significantly improving extreme level accuracy, achieving optimal performance under both MSE and MAE baselines. This study demonstrates that multi-objective loss function optimization effectively improves water level forecasting accuracy, and proposes a systematically designed composite risk-weighted loss strategy to simultaneously alleviate systematic underestimation bias and enhance the fitting performance of extreme water levels for flood risk control. Ablation experiments combined with Bayesian optimized penalty coefficient calibration reveal the independent and synergistic effects of the two penalty terms, offering methodological insights for loss function design in extreme-event hydrological modeling.