Multi-dimensional neural network grey model with delay for intelligent ship trajectory forecasting - Xinping, Xiao; Fangxue, Zhang; Mingyun, Gao (Applied mathematical modelling, 2026)
•Integrates deep learning with grey models for intelligent ship trajectory forecasting.•The Novel model addresses the interactions and time-delay characteristics.•The two-stage method optimizes time-delay estimation and neural networks.•Comprehensive experiments validate superiority across navigation scenarios. Ship trajectory prediction is a key technology for intelligent shipping, playing a crucial role in effective vessel monitoring and future autonomous navigation. Current studies struggle to accurately describe the impact of historical trajectory, heading, and speed on ship movement. To address this issue, this paper proposes a multi-dimensional neural network grey model with delay system structure for forecasting intelligent ship trajectory. This model represents the trajectory prediction system as a set of equations, capturing the interactions between future trajectories and multiple key factors. Specifically, a dynamic time-delay term is introduced in the grey modeling process to reflect the delayed influence of historical trajectories on future movements. For the first time, a neural network is embedded into the traditional system grey prediction model to describe the complex nonlinear relationships between ship motion, heading, and speed. This enhances the model’s nonlinear mapping and generalization capabilities. Additionally, to improve prediction accuracy, an innovative new information recursive least squares algorithm is proposed to estimate time-delay parameters, while neural network parameters are optimized using gradient descent. Finally, the effectiveness of the proposed model is validated through comparisons with three machine learning models, one statistical model, and two grey models. Results demonstrate that novel model exhibits strong robustness and high prediction accuracy, particularly in edge navigation scenarios involving complex nonlinear trajectory data.