Hybrid deep learning method for predicting vertical bending moments at midship

Rapid and accurate prediction of the vertical bending moment at midship is critical for ensuring the safety of ships in severe sea conditions. However, conventional prediction methods are not effective for achieving highly precise and reliable uncertainty quantification. This is primarily attributed to the strong nonlinearity and chaotic nature of marine load signals, which is further complicated by high-frequency disturbances such as wave-induced vibration and slamming events. To overcome these limitations, this study proposes a hybrid deep learning prediction method that integrates variational mode decomposition, a bidirectional gated recurrent unit, and a quantile regression neural network. The three key stages of this method are as follows: First, the original bending-moment signal is decomposed into a set of low-complexity subsequences to mitigate interference induced by the inherent nonlinearity of the signal. Thereafter, the Bidirectional Gated Recurrent Unit (BiGRU) network is used to capture the temporal dependencies of the subsequences for deterministic prediction. Finally, the prediction uncertainty is quantified based on the conjunction and confidence intervals from the Quantile Regression Neural Network (QRNN). The hybrid model was validated in evaluations of a ship model with wave-induced vibration and slamming events. Satisfactory performance was achieved, with an average coefficient of determination of 0.9533 and a coverage rate of 94.22% at the 95% confidence level. These results confirm that the model effectively enables accurate deterministic predictions and effective uncertainty quantification for the vertical bending moment at midship, offering practical support for ship design and real-time operational decisions.