Forecasting and uncertainty estimation of the Shahroud River discharge using long short‐term memory and Kolmogorov–Arnold networks

Forecasting river discharge plays a vital role in water resources management, flood control, and also the safe design of hydraulic structures. Due to the complexity and nonlinearity of hydrological processes, in this research, long short‐term memory (LSTM) and Kolmogorov–Arnold networks (KANs) have been used to predict discharge and estimate uncertainties at the Lowshan hydrometric station on the Shahroud River. The input data, including discharge, temperature, precipitation, wind speed, and relative humidity, were collected monthly over a 50‐year period from 1970 to 2020. The models were trained to predict one step ahead using a lookback window of six previous time steps. A Bayesian approximation was applied to convert the models into probabilistic neural networks capable of providing both mean predictions and uncertainties. Performance of models was evaluated using R 2 , mean absoute error (MAE), root mean square error (RMSE), Kling‐Gupta efficiency (KGE), percent bias (PBIAS), Willmott index (WI), and for quantification of uncertainty, the p ‐factor, d ‐factor, and standard deviation (SD) were used. Mean squared error (MSE) and Gaussian negative log‐likelihood (GNLL) loss functions were used to train the models based on their requirements. Despite the limited dataset, the results indicated that all implemented models delivered acceptable performance. In contrast to MSE‐based models, which capture only epistemic uncertainty, GNLL‐based models were able to represent both aleatory and epistemic uncertainties. Among the four evaluated models, KAN‐GNLL showed the best performance, particularly under Scenario S5 (with a six‐step lookback window), achieving high predictive accuracy with R 2 (0.8533), MAE (5.997), RMSE (8.6692), KGE (0.8575), PBIAS (−6.0659), and WI (0.9578) and reliable uncertainty quantification with p ‐factor (0.9286), d ‐factor (1.0614), and SD (6.1281).