In previous work, the hybrid method has been applied and validated for the numerical simulation of motion in waves and manoeuvre of trimaran (Gong et al., 2020, Gong et al., 2021, Gong et al., 2022b), and here, a brief summary to ensure the paper’s completeness has been presented.

Principal Component Analysis (PCA) is a highly effective statistical method used for dimension reduction, data visualization, and feature extraction. The Pearson correlation coefficient has been employed in calculating the linear correlations between ship motion dynamics and environmental conditions when analyzing ship motion (Zhang et al., 2023). In this paper, visualizing the relation between different distances of wave height will reinforce the understanding of the influence between waves and ship motion.

For two different features, α 1 and α 2 , their correlation coefficient can be computed:

where c o v ( α 1 ,   α 2 ) is the covariance of the two features, v a r ( α 1 ) , v a r ( α 2 ) is the variance of feature α 1 and feature α 2 , respectively. The value of ρ ranges from -1 to 1. The closer the value is to 1, the more positive linear correlation between the two features, and the closer the value is to -1, the more negative linear correlation between the two features. When ρ is 1, it signifies a complete positive linear correlation; when ρ is -1, it signifies a complete negative linear correlation. A ρ value of 0 means there is no linear correlation between the variables.

In this paper, assuming that the time history of one feature is denoted as   X = { x 1 ,   x 2 , ⋯ , x n } , and another feature represent as Y = { y 1 , y 2 , ⋯ y n } . X and Y represent a whole time history of the feature with n nodes, then the correlation coefficient of the two features is (Cohen et al., 2009):

Suppose that each wave height feature is represented as X i = { x 1 i ,   x 2 i , ⋯ ,   x n i } , where x n i represents the n-th node in a time history of the i-th feature. The complete dataset of wave features can be represented as a matrix:

Combined with the Equation 8, the overall covariance S and each element in overall covariance were denoted as follows (Martinez and Kak, 2001):

The standard PCA treats all data points and features equally, and cannot capture the most relevant aspects of the data with complex interdependencies or nonlinear relationships. However, the influences between wave and ship change with the increasing distance (Torsvik, 2009). This paper will employ the WPCA with the Pearson correlation as the weight based on the reference (Xiao et al., 2023). Setting an initial value γ of retained variance can be optimized in the model training process. Assign a weight w i to each feature i, reflecting its importance. Modify the computation of the covariance between features i and j to include weights:

The weighted covariance matrix S ′ is then used in place of the standard covariance matrix for PCA:

The eigenvalues and the eigenmatrix of S ′ are λ and a , they satisfied the following formula:

To adjust the eigenvalues and eigenvectors of S ′ and obtain the edge transformation matrix A, follow these steps:

Then, the principal component matrix can be expressed as:

The retention variance is the ratio of the retained variance to the total variance:

LSTM models, also called Long Short-Term Memory models, are a specialized form of recurrent neural networks (RNNs) specifically designed to handle and learn from time sequential data. Their architecture allows them to effectively capture long-term dependencies, making them particularly suitable for tasks involving the understanding and forecasting of time-based patterns. With gated mechanisms, including input, forget, and output gates, LSTMs effectively manage information flow. In time series prediction, LSTMs excel in capturing complex patterns and long-term dependencies (Rithani et al., 2023), making them suitable for tasks like stock prices and weather forecasting. In maritime applications, LSTMs can predict vessel trajectories, detect anomalies, and monitor vessel operational status, thereby enhancing navigation safety and efficiency. The LSTM unit includes the LSTM cells, each of which includes an input gate, an output gate, and a forget gate, as shown in Figure 3 .

The key principles of each gate in LSTM can be concluded as the following equations (the connotation of each symbol in LSTM cell is shown in Table 2):

Extracting valuable features from the waves and ship motions and reducing risks from the high dimension of features are the main problems in training the neural network. In this section, the specific model structure and algorithmic steps which are used to predict ship motion in this paper will be discussed.

The proposed model will divide the features into two groups. Then, the WPCA algorithm will be employed to extract the main information of waves and the reduce the dimension of the wave feature based on the feature weighting coefficients. This method reduces dimensionality while retaining critical information handles noisy data effectively, and adapts to varying conditions. The optimization process of the hybrid model hyperparameters is divided into external and internal circulation. And the internal and external circulations also form the Double-Circulation (DC) module. The internal circulation repeatedly improves the hyperparameters of the hybrid model based on the metrics of prediction results, while the external circulation optimizes the hyperparameters of the hybrid model based on the retained variance value of WPCA.

The workflow and steps of the proposed ship motion prediction model are shown in Figure 4 . In the first step, the data on motion and wave features will be extracted from the results of the numerical simulation. In the second step, the Pearson coefficient between different distance waves is calculated as the weight of WPCA, and then the dimension of the wave feature is reduced based on the weight. In the third step, the proposed wave features and motion features are combined as the input dataset to train the model. In the fourth step, the hyperparameters will be improved through the internal and external circulation based on the prediction results. The detailed schematic diagram of the proposed model is shown in Figure 5 . Table 3 shows specific hyperparameters of the neural network and the training setting of model architecture. Where is the number of final selected features which will be input to the proposed model, the number of the input features will change with the changes of dimension reduction schemes in this paper. The model is composed of seven layers which contains three LSTM layers, three dropout layers, and a dense output layer. All the layers are connected in the order shown in the Table 3. The inclusion of dropout layers aims to mitigate overfitting by forcing the network to learn more robust features that generalize well to new data. The prediction strategy employed involves using 1000 time steps of selected feature data as input to forecast the same feature data for the next fiftieth or more time step.

Three metrics help evaluate the model’s performance in predicting ship movement in this paper, where R² indicates the proportion of the explained variance, RMSE indicates the typical size of the forecast error, and MAE provides a more interpretable measure of forecast accuracy.

In this section, the data of wave height is selected every five meters from the bow of the trimaran to 20m ahead, as shown in Figure 4 . Five datasets have been generated, each of which contains the data of wave and motion features. The specific conditions of each dataset have been shown in Table 4 .

The total time of the navigation under the condition of 0.35 Fn is 55s, and the data from 0s to 45s has been employed to train the proposed hybrid model, the rest of the data is used for verification. Notably, each time step of the dataset is 0.004s. Figure 6 shows the prediction results of the model trained by the different cases. Table 5 shows R², RMSE, and MAE for predicting the model trained by the different cases.

In Figure 6 , the datasets combining motion and wave features perform well in model training. Comparing the prediction results of the ship’s heave in Figure 6A , the results of Case1 and Case2 are more accurate than those from Case3 to Case5. In Figures 6B, D , there is an apparent bias between the original data and prediction results, suggesting that the introduced feature augmentation fails to uphold robust generalization across various features during model training. It can be seen that the dataset obtained by adding features cannot maintain good generalization for most features when training the model. In addition, the effect of waves at short distances on the ship's motion is greater than that of far waves.

In Table 5 , the bias of R², RMSE, and MAE between these cases shows that waves in distances of 0m and 5m are more valuable in training the model. However, the slight variation in error indicates that these features of different distances cannot be ignored during training.

In general, different distances wave height features have different influences on the model's predictive capability. It can be found that the data set combining waves at short distances (0-5m in this case) performs better than the wave features at long distances. However, it also increases the burden of model learning, and if more wave features are added at one time, the problem of dimensional explosion will be generated.

Due to the redundancy of wave height data at different distances, the feature dimension is too high, which increases the complexity of model training. To alleviate the dimensional disaster of the features and improve the efficiency of model training and performance of the model prediction, the wave feature data will be processed with the WPCA proposed in previous section. Figure 7 shows the Pearson coefficient between different features.

Figure 7A shows the coefficients between wave features, and Figure 7B shows the coefficients between wave and ship motions. The Pearson coefficients between wave features and the ship’s trajectory are close to zero, so those effects can be ignored while calculating the weights before reducing the dimension. Pearson coefficients between other motion features and wave features were calculated respectively, and the expected value of the corresponding feature was taken as the weight of dimension reduction. In WPCA, the feature weights reflect their importance and correlation with the target variable. When a high retained variance (95%) is needed, the model must consider more details and complexity, thus requiring more principal components (WPCA_1, WPCA_2, and WPCA_3). When a lower retained variance (e.g., 85%) is acceptable, the model becomes simpler, requiring only two principal components (WPCA_1 and WPCA_2). The model is highly simplified at very low retained variance levels (below 75%), requiring only one principal component (WPCA_1), indicating that this single component is sufficient to explain the major variance and information.

In this section, the performance of three models will be compared based on the prediction results. The models are the WPCA-DC-LSTM model, PCA-LSTM, and conventional LSTM model. The method of WPCA will generate three groups based on different retained variances. The specific features are shown in Table 6 . A mark “√" indicates the feature data included at input, and the mark “x" indicates that this feature was not included at input.

In Case#2.1, the PCA-LSTM model was employed. Cases from 2.2 to 2.4 utilized the WPCA-DC-LSTM model, while Case#2.5 and Case#2.6 utilized the conventional LSTM model.

Figure 8 shows the results of ship’s motion prediction in irregular waves. The first 45 seconds represent the training data used by the models, followed by the input data from 45 to 49 seconds. The last 6 seconds show the output predictions generated by the models compared against the true values of the prediction target. Table 7 shows the performance metrics of the models on both the training dataset and the validation dataset.

In Figure 8 , the three models have good performance of prediction on the selected dataset. Case#2.2 achieved high precision fit in predictions of heave, roll, pitch, and heading. With the decrease of retained variance, the performance of WPCA-DC-LSTM also decreases slightly. The PCA-LSTM model, which also reduces the wave features to three dimensions, has an apparent bias in motion prediction. The prediction results of the conventional LSTM model without adding wave features also shows poor performance.

In Table 7 , the minimum values of RMSE and MAE appear in the results of dataset Case#2.3. The results indicate that while the prediction accuracy of the WPCA-DC-LSTM model shows certain limitations, particularly in comparison to more advanced or fine-tuned prediction frameworks, it allows for a clearer observation of the impact of different methodological variations. For instance, the performance of Case#2.2, with an R² value of 0.43612, significantly outperforms Case#2.3 (R² = 0.41673), despite their RMSE and MAE values being relatively close. This highlights that the retained variance of 95% in Case#2.2 is more effective in capturing meaningful information than the 85% retained variance in Case#2.3, especially in terms of model fit. Furthermore, the comparison between Case#2.5 and Case#2.6 emphasizes the positive effect of incorporating wave features, as the former shows better performance in all metrics, including a higher R² (0.33715 vs. 0.23079), demonstrating the importance of such features in enhancing prediction accuracy.

Overall, the WPCA-DC-LSTM model demonstrates that higher retained variances (above 85%) are critical for achieving reliable predictions, with performance converging at these levels. While the absolute predictive accuracy is moderate, the model effectively avoids overfitting and dimensionality issues observed in conventional methods. This allows a more controlled and interpretable evaluation of the influence of feature selection, dimensionality reduction, and wave feature inclusion on prediction outcomes. Future work should focus on further optimizing feature weighting before dimensionality reduction and incorporating more wave-related features to enhance the predictive capability for ship motion.

Based on wave height data collected at distances of N (less than 20) m from the ship bow, these data are processed using Weighted Principal Component Analysis (WPCA). The processed wave height data, along with motion data, are then combined to input the hybrid model for predicting the ship’s motion. The specific features of the dataset are shown in Table 8 . The features of wave height and 6DOF will be inputted the model.

The advanced time steps of prediction are 50, 75, 100, 125, and 150. Figure 9 illustrates the comparison between the time histories of predicted ship motion based on various input schemes and the original data. Figure 10 presents three metrics of the models on the selected validation datasets.

In Figure 9 , the model has good prediction effects on the selected dataset when the advanced time step is 50. With the increase in the advanced time step, the discrepancy between the predicted values and the original data is more apparent. In Figure 9A , the performance of the heave prediction is worse than that of other motion features. Especially the discrepancy between the prediction results and original values at the marked troughs and peaks is most affected by the increase of advanced time steps. In Figure 9B , there is a phenomenon of poor fitting to the periodic change data when the advanced time steps are more than 100. The prediction of pitch and heading met the expectation to some extent. The high accuracy of the short-term prediction reflects the ability of the WPCA-DC-LSTM hybrid model to extract the features of the ship’s motions and waves height.

In Figure 10 , the RMSE and MAE achieved the minimum value when the advanced time steps were 50, it means the model is effectively minimizing large prediction errors when the advanced time is 50. Although the loss does not increase with the increase of the advanced time step, the overall trend is higher loss at the high advanced time step.

In general, the predictive capability of the model on the new data set is not too bad when the advanced time below 75, indicating that the proposed model has good ability of generalization. And the advanced time of single heading feature have room for testing. The accuracy of the heading prediction results is higher than the other features, showing that the features extracted from the waves have a more positive influence on the heading in the trained model.

The distance from the ship is reduced from 20 to 15 meters, and the wave height data at 0m,5m,10m, and 15m are taken. Then the wave height data processed by WPCA and motion data will be input into the model together to predict the ship’s motion. The advanced time steps of prediction are 50, 75, 100, 125, and 150. The different comparisons between the time history of the ship motion predicted based on different input database processing schemes and the original data is shown in Figure 11 . Figure 12 shows the three final metrics of the models on the selected validation dataset.

In Figure 11 , the model has good prediction effects on the roll, pitch, and heading when the advanced time step is 50. The accuracy of the 50t-s in Figure 11 is similar to that of the 75t-s in Figure 9 . With the increase in the advanced time step, the discrepancy between predicted values and original data is more apparent. In Figure 11A , the performance of the heave prediction is worse than that of other motion features. Apparent biases can be observed at some low advanced time steps. This means that the model’s ability to process long-term dependencies is declining compared to the results in section 4.1. However, the prediction of the heading also met the expectation.

In Figure 12 , the values of RMSE and MAE at higher advanced time steps are larger than that at low higher times steps. The overall loss of Figure 12 is larger than that of the corresponding advanced time step in Figure 10 . It shows that the contribution of the far wave height features to the prediction cannot be ignored during the process of reducing features dimension. When the wave height data at 20m is reduced, the prediction performance of the model worse overall, indicating that the model obtained the less information with the decrease of wave features.

In light of these observations, it is evident that the WPCA-DC-LSTM model's performance is highly dependent on the inclusion of comprehensive wave height data. The decline in prediction accuracy, particularly for heave at lower advanced time steps, underscores the model’s sensitivity to long-term dependencies. The increased biases and overall loss at higher advanced time steps, as shown in Figures 11 and 12 , suggest that reducing the dimensionality of wave height features, especially those at greater distances like 20m, results in significant information loss. This highlights the critical role that distant wave features play in accurate motion prediction. Therefore, ensuring the retention of key wave height data during feature reduction is crucial for maintaining model performance. Furthermore, enhancing the model's capacity to capture long-term dependencies and incorporating advanced feature extraction methods could further improve its robustness and accuracy in predicting ship motion. Additionally, refining noise reduction techniques will help in preserving essential wave information, thereby enhancing the model's predictive capabilities even with fewer features.

Using wave height data from three points—0 m, 5 m, and 10 m from the ship bow—processed by WPCA, and combining these with motion data, input features were created to predict the ship’s motion. Figure 13 shows the comparison between the original data and the time history of ship motion predicted using different input schemes. Figure 14 shows three metrics of the models on the training dataset and validation dataset.

The trend of the prediction values in Figure 13 closely resembles that in Figures 9 and 11 . However, the main distinction lies in the increasing bias between the predicted values and the original data at advanced time steps, particularly at marked points. In Figure 13D , the prediction of the heading shows significant fluctuations after 40 seconds, with a more pronounced bias compared to Figure 9D

In Figure 14 , both RMSE and MAE values increase at higher advanced time steps, with overall losses exceeding those in Figures 10 and 12 at corresponding time steps. While MAE consistently rises with the time step, RMSE fluctuates around an intermediate value, highlighting its sensitivity to large errors and anomalies. Despite the fluctuations, the model's overall performance stabilizes as wave characteristics diminish. However, this stabilization does not imply improved prediction accuracy. The differing sensitivities of RMSE and MAE to factors such as error distribution, outliers, and skewness further explain these variations. Overall, the results indicate a decline in the model's generalization ability as fewer wave features are extracted.

Comparing predictions across the three datasets, the model's performance at advanced time steps deteriorates as wave features are reduced. Wave data may contain noise, but assigning higher weights to significant features during dimensionality reduction reduces the influence of noisy features. This process enhances data quality and model robustness. However, when key features critical to prediction are removed, the model loses essential information. Furthermore, the complementary relationships among wave features are disrupted, limiting the model's ability to capture complex wave patterns.

Overall, the WPCA-DC-LSTM model demonstrates strong generalization capabilities across new datasets. Although its predictive accuracy declines over time steps for certain motion features, such as heading and pitch, it excels in accurately predicting heave and roll motions. The reduction in wave features highlights the model's ability to capture essential information, but it also underscores the importance of comprehensive feature extraction for maintaining prediction accuracy. By assigning higher weights to significant features during dimensionality reduction, the model effectively mitigates the impact of noise and preserves critical wave patterns, even with fewer features. This capability ensures the model remains robust and adaptable, reinforcing its effectiveness in capturing and generalizing complex wave dynamics.

To further validate the effectiveness of the proposed method, a comparative analysis was conducted against several widely used prediction models, including LSTM, BiLSTM, PSO-BiLSTM (BiLSTM optimized with traditional Particle Swarm Optimization), and ADPSO-BiLSTM (BiLSTM optimized with Adaptive Particle Swarm Optimization) (Cheliotis et al., 2020; Han et al., 2024). The goal of this comparison was to highlight the performance differences in terms of prediction accuracy and the ability to handle complex, high-dimensional wave-ship interaction data.

The LSTM and BiLSTM models were chosen as baseline methods due to their popularity in time-series prediction tasks. PSO-BiLSTM and ADPSO-BiLSTM represent more advanced techniques, incorporating optimization algorithms to enhance BiLSTM's predictive capabilities. These methods are frequently employed in the field to address nonlinear and complex dynamic systems. The training dataset consisted of sequential data combining wave features and motion features. The wave features included wave heights at distances of 0 m, 5 m, 10 m, 15 m, and 20 m from the vessel, while the motion features represented the six degrees of freedom (6DOF) of the ship’s movement (surge, sway, heave, roll, pitch, and yaw). These features were chosen to fully characterize the dynamic interactions between the vessel and the irregular wave conditions.

The comparison was performed on the same dataset, ensuring consistency in input features, training parameters, and evaluation metrics. Results were analyzed based on key performance indicators, including RMSE, MAE, and R².

From the Figure 15 , the comparative evaluation of Model.4.1 to Model.4.5 (LSTM, BiLSTM, PSO-BiLSTM, ADPSO-BiLSTM, and WPCA-LSTM) reveals that WPCA-LSTM (Model.4.5) achieves the highest accuracy in predicting ship motion dynamics across all degrees of freedom. While all models capture the general trends of the original data, WPCA-LSTM consistently exhibits the closest alignment, particularly during steady-state oscillations (t>50 s). This performance highlights the effectiveness of combining WPCA for feature extraction with LSTM-based temporal modeling.

In the transient phase (t=36–50s), WPCA-LSTM outperforms other models by minimizing deviations, handling rapid dynamic changes more effectively than earlier architectures. During the steady-state phase, its predictions demonstrate exceptional stability and precision, as evidenced by reduced deviations and accurate periodic oscillation capture. Across specific degrees of freedom, WPCA-LSTM shows superior performance in yaw, roll, heave, and pitch, particularly in capturing high-frequency and peak oscillations where other models falter.

The progressive improvement from Model.4.1 to Model.4.5 underscores the impact of advanced optimization techniques and architectural refinements. WPCA-LSTM leverages these advancements to provide the most accurate and robust predictions, though minor residuals in the transient phase suggest room for further enhancement through adaptive modeling techniques. Overall, WPCA-LSTM sets a new benchmark for predictive performance in complex ship motion dynamics, offering a promising approach for future applications.

In Table 9 , LSTM and BiLSTM demonstrated reasonable performance but were less capable of capturing the full complexity of the dataset, particularly when dealing with irregular wave conditions. PSO-BiLSTM and ADPSO-BiLSTM achieved improved predictive accuracy compared to the baseline methods. However, their performance gains were limited when the dataset's dimensionality increased due to irregular wave data. The proposed method outperformed the baseline and optimization-based models in certain tasks, particularly in capturing the underlying dynamics of wave-ship interactions. Its performance was also more consistent across different prediction tasks.

This comparative analysis highlights the robustness and adaptability of the proposed approach in addressing the challenges posed by high-dimensional and irregular wave data. While optimization-based methods like PSO-BiLSTM and ADPSO-BiLSTM show promise, the proposed method provides a balanced solution, achieving competitive accuracy while maintaining interpretability and computational efficiency.