Modeling ship maneuvering dynamics presents numerous challenges, including long-term multi-step recursive error accumulation, insufficient generalization under distributed control rates, and high-frequency disturbance amplification effects. Traditional analytical models heavily rely on vessel-specific trials to characterize strongly nonlinear coupling terms and perform parameter identification, making it difficult to balance efficiency and accuracy under complex operating conditions. This paper presents a ship maneuvering-oriented integration of an invertible Koopman representation and a NeuralODE-based continuous-time predictor. The IKN reconstructs strongly coupled state spaces while enhancing representational invertibility, whereas NeuralODE directly fits the control differential equations governing ship maneuvering dynamics and supports continuous-time prediction. Experiments validate multi-rate control performance under ideal and disturbed data conditions, assessing error accumulation and extrapolation stability through long-term multi-step propagation. Evaluations utilize the KVLCC2-type L7 ship model with a 0.25 s sampling interval and a 200 s prediction horizon, validated against a multi-rate control test set. The results indicate that, compared to the baseline neural ODEs model without IKN, the normalized root mean square error (NRMSE) of state quantities decreased by 12.68% on average. In typical operational scenarios such as constant-speed emergency turns and variable-speed sine sweep maneuvers, the average state NRMSE was 7.96% lower than the LSTM model and 53.85% lower than the IKN–Koopman operator network. Noise experiments demonstrated that when introducing simulated sensor noise at 5%, 10%, and 20% into the dataset, the average state NRMSE remained at 5.98%, 8.24%, and 10.06%, respectively. This confirms the method’s stable prediction performance under varying noise intensities.
Zhang et al. (2026) studied this question.