Use of Machine Learning for Estimation of Wave Added Resistance and Its Application in Ship Performance Analysis

Several initiatives are being developed to reduce greenhouse gas emissions and to increase ship operational efficiency. These include the implementation of various measures such as operational and logistical optimization, hull and propeller maintenance, utilization of advanced hull coatings, integration of innovative energy efficiency technologies, and the adoption of alternative fuels. In pursuit of this objective, the monitoring of ship operational performance has emerged as an important and effective measure. Monitoring the performance of ships while navigating in actual sea conditions is dependent upon a multitude of factors, including vessel condition, weather situation, and loading condition. One of the most critical quantities that should be computed for performance monitoring is wave added resistance. Accurate estimation of this quantity is a demanding task since it is highly dependent on the geometrical details of the hull and the ship motions.

Currently, the added resistance due to waves can be estimated in numerous ways. These include tank model testing, numerical simulations [1–3] such as computational fluid dynamics or potential-flow models [4], and semi-empirical [5] or empirical formulas [6]. In spite of the fact that experimental measurements and numerical calculations are employed during the ship design phase, these methods usually cannot be used practically for estimation of wave added resistance in performance monitoring. First, it is extremely time demanding, and next it requires a complete knowledge of the hull geometry, which in the majority of cases is not available. It is quite common for shipping companies to only have the IMO number of a vessel for which they would like to conduct performance monitoring. This piece of information provides only a limited amount of general geometrical data like length, beam, draft, and block coefficient. Therefore, the usual current practice is to employ simple empirical or semi-empirical formulations, for example Refs. [7,8], for estimating the wave added resistance. These methods are computationally fast and vary in terms of their generality and reliability.

In recent years, the utilization of machine learning methodologies and deep neural networks (DNNs) has emerged as a novel approach for predicting ship performance and estimating added resistance in the maritime industry. These models are computationally efficient, and if they are created based on reliable data sources, they could become a very promising tool for estimation of wave added resistance in performance monitoring or even in the earlier stages of design to assist getting a preliminary estimate of the total ship resistance. However, one disadvantage of using DNNs is that they depend strongly on the quality of the training data. This means that poor quality training data lead to an inaccurate model and/or a nongeneralizable model. Previous studies related to wave added resistance prediction based on DNNs have utilized either experimental data or semi-empirical formulas and only for head waves. In what follows, we present a brief review of these previous studies.

A two-layer perceptron model was employed by Ref. [9] (excluding the input layer) to predict the added resistance transfer function of ships encountering head waves. The model utilized inputs such as waterplane area, waterplane coefficient, ship speed, and incident wave frequency, while the output was nondimensional added resistance coefficient. The model was trained by numerical simulations. In another study, the inputs were expanded [10] to include the ships’ length, beam, draft, block coefficient, Froude number, and the ratio of wavelength to ship length. The training dataset used by Cepowski was obtained from the published experimental data. In a follow-up study, the issue of determining added resistance during the design phase for ships encountering head waves was addressed in Ref. [11]. This study involved the development of a collection of five DNNs aimed at predicting the wave added resistance. These DNNs were trained using fundamental design parameters including ship length, beam, draft, block coefficient, and Froude number. Since the training dataset was derived from the characteristics of standard hulls, the accuracy of the developed DNNs model was limited to such typical vessels.

A deep feed-forward neural network approach was introduced in Ref. [12] to efficiently estimate the added resistance of ships in head seas. The training datasets used in this study were generated using semi-empirical formulas. The input layer of the network comprised several variables, including the ship’s length, beam, draft, prismatic coefficient, angle of waterline bow area, Froude number, longitudinal mass radius of gyration, and nondenationalized incident regular wavelength, all of which were normalized by the ship’s length. However, the model demonstrated limited capacity in effectively handling data that fell outside the range of the training dataset. The Holtrop–Mennen method [13] as a primary basis for evaluating the calm water resistance of ships was employed in Ref. [14] to calculate the added resistance resulting from weather conditions, and the study incorporates a machine learning algorithm named random forest.

In this article, a machine learning model is presented for fast and reliable estimation of wave added resistance. Motivated by the importance of data quality for training a DNN model, in the present work, we obtain the training dataset using two-dimensional (2D) strip theory calculations, i.e., through simulated data. The calculations encompass a range of operational conditions, including varying relative heading angles from head to following seas, and a large number of hull geometries. On the basis of these computations, we have generated a rich database of added resistance transfer functions. Next, the database is partitioned into training (75%) and test (25%) subsets. The DNN model undergoes training using the training subset, employing a range of deep learning hyperparameter optimization techniques to enhance its predictive accuracy. Subsequently, the model’s performance is evaluated on the test subset, comprising unseen data to assess its accuracy. This evaluation demonstrates that the DNN model’s predictions closely align with the direct numerical calculations performed on the test dataset. It is important to note that employing simulated data for the training dataset ensures an unbiased database, which consequently facilitates the application of the DNN model to diverse ship types. In the end, the developed DNN model is employed together with real in-service data for assessing required propulsion power and biofouling effects.

The structure of this article is as follows. In Sec. 2, we describe the methodologies that have been employed to develop the DNN model for computing wave added resistance. In Sec. 3, the outline of the procedures used for the vessel performance analysis is briefly reviewed. In Sec. 3, the results for the performance analysis of three bulk carriers are also shown. Finally, in Sec. 4, we present the discussion and conclusions.

We develop a machine learning model for computation of wave added resistance in the performance analysis. Training of the DNN model for added resistance is performed using a dataset, which is generated by strip theory calculations by the Technical University of Denmark (DTU) Strip Theory Solver package [15], henceforth named DTU Solver in this article. This solver is an implementation of Salvesen–Tuck–Faltinsen strip theory [16] and provides wave added resistance computations based on both a modified version of Salvesen’s formulation and Maruo’s method. Full details about these methods can be seen in Refs. [17–20]. The DNN model is based on the modified version of Salvesen’s method, henceforth named only Salvesen’s method.

Figure 1 shows the structure of neural network with the input layer, the output layer, and the hidden layers for this study. For our model, the deep neural network gets the following input variables and ranges:

• Beam ratio (⁠$B/Lpp∈[0.125,0.25]$⁠)

• Draft ratio (⁠$D/Lpp∈[0.025,0.125]$⁠)

• Froude number (Fr $=V/gLpp∈[0,0.3]$⁠)

• Wave heading angle (⁠$β∈[0,180∘]$⁠)

• Slenderness ratio $(f∈[3.42,7.38])$

• Nondimensional wave frequency (⁠$ω¯=ωLpp/g∈[1.45,$$4.58]$⁠)

The generated database presented in Table 2 is partitioned into training (75%) and test (25%) subsets. Here, $B$⁠, $Lpp$⁠, $D$⁠, and $f$ are beam, length between perpendiculars, draft, and slenderness ratio of the ship, respectively. The slenderness ratio is given by $f=Lpp/∇1/3$⁠, where $∇$ is the ship displacement and the block coefficient is $Cb=∇/(LppBD)$⁠. $β$ is the wave heading angle relative to the ship, and $ω¯=ωLpp/g$ is the nondimensional incident wave frequency. $g$ is the gravitational acceleration, and $V$ is the forward speed of the vessel. The output is the wave added resistance coefficient (nondimensional added wave resistance) denoted by $Cw¯$⁠.

The machine learning framework for developing the deep neural network in this work is the Tensorflow platform [23]. In machine learning, there are several settings called hyperparameters to control the behavior of the learning algorithm, and these hyperparameters are not adjusted by the learning algorithms automatically [24]. The initial step is to implement fivefold cross validation [24] to evaluate a preliminary number of hidden layers and a number of neurons per layer for a variety of activation functions (⁠$ReLU$⁠, $SELU$⁠, $ELU$⁠, and $GELU$⁠). The result of fivefold cross validation assists in initiating the preliminary model in terms of a mean cross validation score for five different validation (unseen) datasets and helps to prevent overfitting and to increase the generalization ability of the model [25]. The top three initial model structures are shown in Table 3, where $R2¯$ is the mean $R2$ score among five distinct validation datasets. It should be noted that these models are not the final models. In the cross validation step and sensitivity study, $GELU$ was found to be the best activation function because it is a smooth approximation to the rectifier function. It has been shown to outperform other activation functions in various tasks, as discussed in Ref. [26]. Also, $GELU$ can handle dead states caused by dying neurons or nearly vanishing gradients [27]. Moreover, $GELU$ can cover dropout regularization. Dropout regularization is a regularization technique where randomly selected neurons are ignored during training.

The best preliminary models were next trained for more epochs and using the early stopping method [28] followed by fivefold cross validation to prevent overfitting. Early stopping is done by stopping the training once the performance on a validation set gets worse [29]. Hyperparameters were tuned in the following order:

1. Optimizers (Adam, RMSprop, Adadelta, Adagrad, Adamax)

2. Learning rate (⁠$α$⁠)

3. Numbers of layers, neurons, and batch size

The optimizer is an important algorithm in a neural network that adjusts the weights and learning rate of a model. It assists in mitigating the overall loss and boosts the accuracy of the model. Loss degradation during the training should be smooth and less noisy. The less noisy the loss degradation, the more robust the neural network becomes in terms of performance and accuracy. To find the optimal optimizer, some available optimizers have been tested on training and validation datasets. The validation dataset is a random 25% sample of the full dataset. Figures 23 and 24 show that the $Adamax$ optimizer achieves lower RMSE with smooth and steady loss degradation for both training and validation datasets. $Adamax$ is an extension to the $Adam$ version of gradient descent that generalizes the approach to the infinity norm (max), and $Adamax$ has the ability to adjust the learning rate based on data characteristics [31]. $Adamax$ adjusts the learning rate based on data characteristics, which can lead to better model performance [32].

In the next phase, the tuning is implemented for a range of learning rates $α=[0.00001,0.0001,0.001,0.01,0.1]$⁠, in order to find its optimal value. The learning rate is a hyperparameter that determines how much the weights in the network will change in each update. If the learning rate is too small, the training process can become slow and might get stuck in a suboptimal solution [33]. If the learning rate is too large, the training might not converge or even diverge. As shown in Figs. 25 and 26, the optimal learning rate for the DNN model is 0.001, as the loss degradation is steady and can achieve a low RMSE value. Selecting an excessively small learning rate can lead to a prolonged training process, which may need to be terminated prematurely. Conversely, choosing an excessively large learning rate can result in rapid and unstable training [34].

For finding the optimal number of neurons, layers, and batch size, many trials in a sensitivity study have been implemented. If the number of layers and neurons are large, then the model will be prone to overfitting. This means that the model has a memorization ability rather than generalization ability, which makes the model predict well on training data (seen data), but unable to predict unseen data. The results of testing the final DNN model on the unseen dataset (different hull shapes) is shown in Sec. 2.3.

In order to validate the accuracy of the DNN model, a comparison is made between the added resistance predicted by machine learning and the one that is obtained by direct computations using the DTU Solver. Computations are carried out for two hull geometries, which were not involved in the training process of the DNN model. Their main particulars are presented in Tables 4 and 5. Note that for both geometries, $Cb≥0.8$⁠. A plot of these hull lines is also shown in Fig. 2.

1. The Sea Horse hull is a bulk carrier. The result for the model correlation is shown in Fig. 3. With a correlation coefficient of $R2=0.9993$ and an $RMSE=0.0615$⁠, the DNN model exhibits an overall high degree of accuracy. In Fig. 4, the DNN model results are compared with direct calculations by the DTU Solver. As can be seen, a remarkable agreement has been achieved for all Froude numbers, heading angles, and wave frequencies, which is shown in Fig. 4.

2. The Torm Lilly hull is a product tanker. The findings related to the model’s correlation are shown in Fig. 5. For this case, the correlation coefficient $R2=0.9979$ and the $RMSE=0.0909$⁠, which indicates a notably high level of accuracy in the DNN model. Furthermore, Fig. 6 illustrates a comparison between the results generated by the DNN model and direct computations performed by the DTU Solver. As can be seen, the DNN model is able to predict the computed added resistance with a remarkably high accuracy, which can be observed in Fig. 6.

For training the DNN model, we utilized an NVIDIA A100 80GB PCIe, provided by The Technical University of Denmark (DTU) HPC support [30]. This choice was motivated by the time-intensive nature of the training, particularly during hyperparameter tuning, which took approximately 14 h when leveraging the parallel processing capabilities of the DTU HPC cluster. The high-performance computing resources from DTU HPC significantly accelerated the process, making training faster and more efficient compared to conventional methods. Once the DNN model was finalized and fully trained, it can now operate on a standard computing platform with a specification of AMD Ryzen 7 PRO 4750U with Radeon Graphics at 1.70 GHz, completing inference tasks within approximately 4 seconds. This level of computational efficiency represents a substantial improvement over the direct calculations, which also require detailed knowledge of the hull geometry.

In this section, the DNN model is compared with experimental data and with the numerical results using the SNNM method developed by Liu and Papanikolaou [5]. For more information regarding this semi-empirical method, refer to Ref. [35]. The experimental measurements are for two hull geometries that were not part of the training dataset. The first is a tanker ship called the SNU Tanker [36] with the main particulars $Lpp=323(m)$⁠, $B=60(m)$⁠, $D=21(m)$⁠, and $Cb=0.83$⁠. The other ship is a bulk carrier called SB84 [37] with the main particulars $Lpp=178(m)$⁠, $B=32.26(m)$⁠, $D=11.57(m)$⁠, and $Cb=0.84$⁠. The results are shown in Fig. 7. For beam to head seas conditions, a relatively good agreement with both the SNNM method and the measurements can be seen. However, for the following seas cases, the disagreement between our method and the experimental data is considerable. Both measurement and computation of wave added resistance in following seas, and also in relatively short-waves, are extremely challenging and subject to large uncertainty. The accurate estimation of wave added resistance in these cases is the subject of intense ongoing research, from both a computational and an experimental perspective. A common computational approach in the high-frequency range is to replace direct calculations with the asymptotic theory, as discussed in Ref. [35]. We have not pursued that approach here, however, as the model appears to be sufficiently accurate for the desired application.

A decision to schedule dry docking and the related hull and propeller cleaning is directly dependent on a proper and reliable performance analysis. In this section, we apply the developed DNN model for predication of wave added resistance inside a vessel performance framework.

We conduct the performance analysis based on a comparison between the total shaft power reported by the ship during operation and the power calculated based on the reported operational conditions with respect to calm water resistance plus wind and wave added resistance. Note that in this analysis, shallow water resistance and other (minor) unsteady resistance components have not been considered. In the following sections, first, a brief review of the methodologies and the formulations, which we have used for the performance analysis, is presented.

Here, $CF$ is the frictional resistance coefficient, $CA$ is the incremental resistance coefficient, $CAA$ is the air resistance coefficient, and $CR$ is the residual resistance coefficient. For the detailed calculation of these coefficient, refer to Ref. [40].

The wave added resistance is defined as the mean of the unsteady force caused by the encountered waves and swells on the ship in the longitudinal direction. In principal, the estimation of the wave added resistance requires a detailed knowledge of the hull geometry in order to capture the related hydrodynamic features of the problem. However, as mentioned in Sec. 1, we obtain the added resistance curves using the DNN model developed in Sec. 2. In this section, we describe the formulation used for computing the total mean wave added resistance $R¯Wave$⁠.

For the power and fuel consumption prediction, the considered operational data (noon data and weather data) are related to three bulk carriers in three different size classes, Handysize, Supramax, and Panamax, from the Ultrabulk fleet. The ship main particulars are presented in Table 6.

The full dataset contains the ship main particulars, noon reports, and the related hindcast weather data to the noon reports provided by WNI (Weathernews Inc.)³ from 2020 to 2022. Data handling of the noon reports is a critical part of the fleet performance analysis since there might be outliers (invalid values) and NaN (not a number) values in the reports sent by the ship crew. Therefore, for data handling of noon reports, the following filtering criteria have been considered.

1. $Mean draft>Db−1(m)$

2. $Mean draft<Ds+1(m)$

3. $VOG>0(Knots)$

4. $VTW>5(Knots)$

5. $Main engine power>0$

6. $Main engine power<1.1MCR ( KW)$

Here, $Db$ is the ballast draft, $Ds$ is the scantling draft, $VOG$ is the speed over ground, $VTW$ is the speed through water, and maximum continuous rating (MCR) is the engine maximum continuous rating. The final features needed from the noon reports for ship power prediction and performance evaluation are as follows: date and time, mean draft, main engine power, ship speed over ground, ship speed through water, ship course (heading), significant wave height, primary wave direction, primary wave period obtained from hindcast data, true wind speed, true wind direction, and current factor combined with hindcast weather data. Here, the current factor refers to the ocean current speed and direction relative to the ship.

In the noon reports, there are two reported speeds: speed over ground (⁠$VOG$⁠) calculated from the GPS position and speed through water (⁠$VTW$⁠) from the speed log onboard the vessel. Measurements by an installed log speed sensor on the ship hull can be easily disturbed and influenced by surface fouling. A reliable measurement is thus highly dependent on a regular calibration and proper maintenance of these sensors. Note that the reliability of the Doppler velocity logs has been investigated (and questioned) recently in several studies. See, for example, Ref. [45–48]. Therefore, in the context of the performance framework in this article, the ship speed through water has been determined by combining ship speed over ground and the relative current factor obtained from the hindcast data. In this study, a current-corrected speed is used for the analysis because it is a better estimate of the speed through water than the logged speed [49]. This corrected speed through water is obtained by subtracting the current speed provided by the hindcast data from the speed over ground (⁠$VOG$⁠). Figure 8 shows the distribution of current speed for the Supramax vessel. In addition, an example of the distribution of primary wave period (⁠$TW$⁠) is presented in Fig. 9. As can be seen, most of the wave energy encountered by this vessel is in the short wave range, since the length of the vessel is 195 m and the most frequent wavelength during sailing is 87.87 m, which is small in comparison to the length of the vessel.

The ITTC formula (10) has been used for computation of wind added resistance. The weather data provide the true wind speed and the true wind direction. So, to calculate the wind added resistance, these values must be converted to wind speed and direction relative to the vessel. A definition of the true and the relative wind is shown in Fig. 10.

In the case studies presented in this article, the relative wind speed and direction were calculated based on true wind speed and direction from hindcast data using the ship speed over ground (⁠$VOG$⁠) and the ship heading angle from noon reports. Then, by applying the ITTC formula, the wind added resistance was calculated for each noon report. An example of this calculation is presented in Fig. 11. Note that in the plot, the relative wind direction of $180$ deg signifies a wind coming from behind, and $0$ deg corresponds to head-wind conditions.

The mean wave added resistance based on the DNN model and the operational report sent every 24 h is shown in Fig. 12. In this plot, the wave direction $180$ deg indicates head seas, and $0$ deg indicates the following seas. As can be seen, for waves coming from beam to head seas, the wave added resistance is positive and opposite to the sailing direction; however, the mean wave added resistance is negative in following seas, which is consistent with Ref. [15] and also found in other studies analyzing full-scale data, e.g., the studies by Nielsen et al. [50] and Mittendorf et al. [51].

The calm water resistance can be calculated according to Ref. (7) using the ship speed through water (⁠$VTW$⁠) and the total resistance coefficient $CT$⁠. In Fig. 13, the computed calm water resistance is plotted with respect to the speed through water. Also, the ratio of the combined wind and wave added resistance to the calm water resistance with respect to the significant wave height (⁠$Hs$⁠) is presented in Fig. 14. The ratio of the combined wave and wind added resistance to the total calm water resistance is plotted with respect to the speed through water $VTW$ in Fig. 15. All of the aforementioned calculations have been performed for other ships (Handysize and Panamax), and similar results have been obtained.

The total estimated resistance experienced by the ship is the sum of the calm water resistance and the added resistance due to wind and waves, noticing that shallow water resistance, fouling resistance, and increased rudder/steering resistance are not considered in this study. After estimation of the total resistance, the main engine power can be predicted based on the procedure implemented and described in the previous sections. In Fig. 16, an example of the computed delivered power of the main engine is presented. Each point in the figure represents the estimated power based on the velocity and draft from the noon reports, together with the hindcast environmental conditions. The two lines show the main engine power computed based on only the calm water resistance for a clean hull in drafts 12 (m) and 5 (m) in the speed interval $VTW∈[7.5,15.5]Knots$⁠. From the figure, the magnitude of the combined wind and wave added resistance relative to the calm water resistance can also be observed.

In Fig. 17, the biofouling analysis for the vessels are presented. The analysis has been carried out according to Sec. 6. It can be observed that after long port stays (shaded areas), the percentage power increase is increasing, which is likely to be due to fouling of the hull and propeller. Moreover, the percentage power increase drops after cleaning events, which indicates that the fouling has been removed during the cleaning, and accordingly, the ship requires less power for sailing. Those occasions, where no improvement is observed after dry docking, could indicate either an insufficient cleaning process or serious errors in the noon reports by the crew. Identification of such suspicious events will assist the ship operator in taking the necessary measures so as to rectify the noon reporting or to perform more efficient hull and propeller cleanings.

It is important to mention that the deviation between the reported and the estimated power could also be ascribed to other factors such as a mismatch between the true environmental conditions and the hindcast data, or errors in the added resistance predictions. The difference in the reported main engine power and the estimated delivered power to a certain extent could also be due to main engine problems. This has not been considered in this study. The reported main engine power from the noon reports is only available at a very low sampling rate (e.g., every 24 h). Therefore, it should be more reliable to perform main engine performance monitoring instead using readings from the torque meters.

Referring to Fig. 17, the deviation between the reported and the estimated power may be an indicator of the fouling growth on the hull and propeller. This consequently leads to the required measures to schedule dry docking. On the other hand, this deviation could be due to severe misreporting of the power by the ship crew. This is particularly noticeable right after a hull and propeller cleaning, when the operator expects a considerable reduction in the fuel consumption. In this case, the performance analysis triggers prompt action to investigate the source of the potential errors in crew reporting. Low-quality hull and propeller cleaning could also be responsible and could be discovered by this type of performance analysis. In any of these situations, the developed computational framework provides a valuable tool for a ship operator to increase the performance efficiency of the fleet.

It is important to mention that the performance computations presented in this article are subject to several uncertainties. This is a common issue in almost every vessel performance calculation tool, which makes proper decision-making rather difficult. These uncertainties have different sources, and each should be tackled separately through its own methodology. Except for dealing with the uncertainty related to the wave added resistance, our objective in this work has not been to deal with any other underlying uncertainties in the performance analysis. The reduction of uncertainty due to the wave added resistance has already been investigated in our previous study in Ref. [52]. In that work, we have illustrated and quantified how a curve-based added resistance computation (like the one utilized here) will reduce the uncertainty of power calculations in comparison to the case where the wave added resistance is estimated using simplified empirical formulations.

Although the developed DNN model demonstrates good performance in predicting wave added resistance, several limitations need to be considered. One key limitation is that the model has been trained on a dataset generated through numerical simulations using strip theory, which, while efficient, inherently simplifies the complex physical phenomena of wave–ship interactions. This means that the model’s accuracy is bound by the assumptions and limitations of the strip-theory calculations, potentially leading to reduced precision when applied to conditions or hull forms that may violate these assumptions.

Another concern is the dependency on the quality and scope of the simulated dataset. If the numerical calculations do not encompass the full range of operational conditions or ship designs, the DNN’s generalizability may be compromised, especially for hulls or sea states outside the range covered by the presented simulations. Additionally, the DNN model’s “black box” nature makes it challenging to interpret how it arrives at certain predictions, which can be problematic in situations where model transparency is required for decision-making or regulatory purposes. Finally, while DNNs provide computational efficiency and scalability, real-world factors such as sensor inaccuracies, environmental uncertainties, and errors in input data could affect the model’s reliability when used in practice.

Incorporating a sensitivity analysis provided valuable insights into the robustness of our DNN model, particularly regarding key input parameters such as block coefficient (⁠$CB$⁠) and slenderness, which play a significant role in distinguishing vessel types and influencing wave-added resistance predictions. We found that the block coefficient is a critical parameter for model accuracy. Consequently, our model training focused on ships with higher block coefficients (⁠$Cb≥0.8$⁠), where slenderness emerged as another essential parameter for training the DNN model. Due to the limited scope and time frame of this project, which was conducted as part of a Master’s thesis, we were unable to perform a detailed sensitivity analysis. We recognize its importance and plan to incorporate this analysis in future work.

Finally, we emphasize that any decision-making for ship performance using the presented computational framework should be conducted with the reservation that there are other sources of uncertainties, which have not been addressed yet. These include uncertainties due to crew reports, hindcast data, steering/rudder/shallow water resistance, nonlinearity in ship response, and the model for estimating wind forces. In case of mitigation of the uncertainties related to the wave spectrum, utilizing the wave buoy analogy in data collection (and for immediate decision-making support) holds promise in improving the accuracy of sea state information. See, for example, Refs. [53,54].

In the domain of deep learning, the efficacy of a model is intricately linked to decisions made during its construction and training phases. Two critical factors that exert substantial influence on the performance of deep learning models are the careful tuning of hyperparameters and the thoughtful selection of a training dataset. This research employs a diverse array of hyperparameter tuning methods to enhance the accuracy and robustness of machine learning models. In contrast to the previous studies that predominantly have relied on experimental data for model training, this research utilizes a training dataset obtained using numerical calculations. This allows for a more generalized approach, avoiding limitations associated with specific ships and operational conditions encountered in the experimental data. In this work, the local minima problem has been solved by studying different optimizers and learning rates. It is suggested that the training dataset should be supplemented with more hull points and input features to get even better agreement with the underlying numerical model.

The main objective of this study was to develop a deep neural network model for the prediction of wave added resistance. The training dataset was generated using strip theory calculations over a wide range of ship geometries, Froude numbers, wave heading angles, and frequencies. The resultant DNN model provides fast estimation of the wave added resistance using only the bulk geometrical data and the ship operational conditions. At the same time, from a hydrodynamic point of view, the developed added resistance model is more reliable than the simplified empirical formulations often adopted by consulting companies. This is due to the fact that the DNN model is based on actual wave added resistance calculations that are performed using real and representative ship hull geometries. We have validated the developed DNN model and illustrated that it is in excellent agreement with direct numerical computations by the DTU Solver.

The secondary objective of the project was to determine the expected main engine delivered power of ships in operation and compare it with noon reports. This has been performed by integrating the developed neural network model into a ship performance computational framework. Equation (6) has been applied to determine the deviations between the reported and the estimated power, with some results presented in Fig. 17. The calculated power can assist a ship operator to evaluate the performance of the hull and propeller, which is critical for the energy rating of the vessel and ensuring cost-effective voyages for a shipping company.

The developed deep learning model for estimating the wave added resistance, together with a method for prediction of the ship propulsive power, like the one that is used in this article, can be the basis for further development of a more accurate ship performance monitoring tool. Such a tool can be used to improve the performance of existing ships by maximizing the hull and propeller performance through effective maintenance activities. Furthermore, the tool can be used to quantify the improvement of any new energy saving devices retrofitted on the ship or to investigate the effect of applying any advanced antifouling coating. These efforts from a ship operator’s side directly contribute to the reduction of greenhouse gas emissions. In addition, these measures will assist the ship operator in improving the fleet’s environmental ratings such as the energy efficiency operational indicator and carbon intensity indicator.

We would like to express our sincere gratitude to all those who contributed to the completion of this research. We extend our appreciation to Ultrabulk A/S and specifically Claus Andersen; Head of S&P and Long Term Chartering at Ultrabulk for his valuable insights. In addition, Harry B. Bingham and Mostafa Amini-Afshar are very grateful to Den Danske Maritime Fond (No. 2022-017), Orient’s Fond (No. 2022), and ShippingLab II project for the granted financial support.

There are no conflicts of interest.

The data and information that support the findings of this article are freely available.⁴