Ship trajectory prediction via a transformer-based model by considering spatial-temporal dependency

With the rapid development of global maritime trade, ship trajectory prediction plays an increasingly important role in maritime safety, efficiency optimization, and the development of green shipping. However, the complexity of the marine environment, multi-factor influences, and automatic identification system (AIS) data quality issues pose significant challenges to trajectory prediction. This study proposes a ship trajectory prediction model based on the Crossformer architecture comprising three core components: Dimension-Segment-Wise embedding, Two-Stage Attention layer, and Hierarchical Encoder-Decoder structure, which efficiently captures spatiotemporal dependencies in ship movement patterns. Through experiments on public AIS datasets, we validate the model using two navigation scenarios (complex turning and smooth sailing) and conducted comprehensive comparisons with traditional models such as gated recurrent unit (GRU), long short-term memory (LSTM), and temporal graph convolutional network (TGCN). Experimental results demonstrate that Crossformer significantly outperforms the comparative models across multiple evaluation metrics including average Euclidean distance error (ADE), mean square error (MSE), root mean square error (RMSE), and mean absolute error (MAE), reducing average error by over 60% in complex scenarios and up to 70% in smooth scenarios. For Case 1, Crossformer achieved the lowest values across metrics with ADE of 2.35 × 10-2, MSE of 7.00 × 10-4, RMSE of 2.58 × 10-2, and MAE of 2.35 × 10-2, substantially outperforming GRU, LSTM, and TGCN models. For Case 2, Crossformer similarly excelled with an ADE of 1.64 × 10-2, MSE of 4.00 × 10-4, RMSE of 2.06 × 10-2, and MAE of 1.64 × 10-2. The model maintains low error levels in predicting both latitude and longitude dimensions, exhibiting excellent multi-dimensional prediction capability and robustness. This research not only provides a high-precision solution for ship trajectory prediction but also establishes an important technical foundation for intelligent ship scheduling, maritime traffic management, and navigation safety assurance.