Data and knowledge hybrid driven ship navigational speed prediction method

Through a hybrid drive method combining meteorological, AIS and industry knowledge data, the shortcomings of existing ship speed prediction methods are solved, and efficient and low-cost speed prediction is achieved, which is suitable for the intelligent process of shipping.

CN120296573AActive Publication Date: 2025-07-11无锡九方科技有限公司
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Patent Information

Application Number
CN202411962096.X
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2024-12-30
Publication Date
2025-07-11
Estimated Expiration
2044-12-30

AI Technical Summary

Technical Problem

The existing ship speed prediction method relies on empirical statistics, ignores the complex interaction of environmental factors, lacks real-time and dynamic adjustment capabilities, and has limited scope of application. It is difficult for purely data-driven models to obtain sufficient amount of high-quality data in shipping business, which is expensive.

Method used

The hybrid-driven method of data and knowledge is adopted, combined with meteorological data, AIS data, ship static data and industry knowledge data, and the ship speed prediction is carried out by building a hybrid-driven learning model, using industry experience formulas and data science processing technology, anomaly data is eliminated and normalized, and the model training is used to minimize the loss function optimization parameters.

Benefits of technology

It significantly improves the accuracy of speed prediction, reduces construction costs, and can achieve long-term accurate predictions in real shipping business, effectively deal with small sample problems, reduce economic losses and ensure crew safety.

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Abstract

The invention belongs to the technical field of artificial intelligence, and particularly relates to a data and knowledge hybrid-driven ship navigational speed prediction method. According to the scheme, the industry knowledge data is used as one of the data sets, and the ship navigational speed is predicted by adopting a'data + knowledge 'hybrid driving training mode, so that the problem of small samples in shipping services is effectively solved, the navigational speed prediction accuracy is remarkably improved, and the construction cost is reduced.
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Description

Technical Field

[0001] The present invention belongs to the technical field of artificial intelligence, and specifically relates to a method for predicting ship speed driven by a hybrid of data and knowledge. Background Art

[0002] The Automatic Identification System (AIS) transmits detailed ship information through radio, including ship name, position, course, speed, etc., providing a solid foundation for ship monitoring, collision avoidance, route optimization, and behavior analysis. Combining AIS data with deep learning models can accurately predict ship speed, anticipate its spatio-temporal position in advance, and effectively avoid potential risks, which not only improves navigation safety but also promotes the process of shipping intelligence.

[0003] In current operations, ship speed prediction mainly relies on empirical statistical methods. By analyzing the characteristic laws based on historical data, a statistical model is constructed for prediction. However, this method ignores the complex interactions of environmental factors, lacks real-time performance and dynamic adjustment capabilities, and has limited applicability. With the rapid development of artificial intelligence technology, machine learning models have demonstrated excellent performance in many fields due to their high efficiency and accuracy. However, pure data-driven network models often rely on a large amount of labeled data for training to ensure their accuracy and good generalization ability. In the actual application of the shipping business, it is often extremely difficult and costly to obtain a sufficient amount of high-quality data. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for predicting ship speed driven by a hybrid of data and knowledge, which requires fewer samples and has low costs.

[0005] To solve the above problems, the following technical solutions are provided: The method for predicting ship speed driven by a hybrid of data and knowledge of the present invention is characterized by including the following steps: First, collect the original data and preprocess the original data to obtain standardized data ; wherein, the original data includes meteorological data, AIS data, ship static data, and industry knowledge data; Second, divide all the data into multiple training sets and test sets. Both the training sets and test sets contain feature data for inputting into the learning model and label data for verifying the prediction results of the learning model; Third, input the training sets and test sets into the learning model to train the learning model until the optimal parameters are found, and save the optimal parameters; Fourth, input a new test set containing historical data and industry knowledge data into the prediction model to realize the prediction of ship speed.

[0006] The industry knowledge data is obtained by calculating an empirical formula; the empirical formula includes the formula of the Russian Central Shipping Research Institute, the formula of Qingdao Meteorological Navigation Research Consortium, the formula of the National Meteorological Center of the State Meteorological Administration and / or a semi-empirical formula; wherein the semi-empirical formula is: , in, is the actual speed of the ship in the waves; is the speed of the ship in still water; According to the wind and wave direction and wind level Varying stall direction factor; is the square coefficient, loading condition and Froude number The stall correction factor varies with the change; For ship type and wind level and The displacement of the ship in units Related ship type factors.

[0007] In the first step, the process of preprocessing the original data is as follows: S1 removes abnormal data. The specific rules are: (1) If all meteorological data in a data record are displayed as 0, the data will be eliminated; (2) If the speed is less than 4 knots or greater than 20 knots, the data shall be discarded; (3) Wind level Less than 4, if the ship speed during normal navigation is between 4 and 5 knots, the data will be eliminated; (4) If the ship charter speed is 0, the data will be eliminated; (5) Wind level When it is equal to 0, the data with the difference between the ship speed and the charter speed greater than 8 will be eliminated.

[0008] S2 encoding process: Use 0-1 coding to convert the ship loading status: in, For no load, is fully loaded, that is, if the ship is empty, the code is 0; if the ship is fully loaded, the code is 1; S3 normalization processing: The data is normalized by normalization method and converted to data in the range of [0, 1]. .

[0009] The normalization is the Min - Max normalization method, which transforms the data into the interval [0, 1] through the following formula: , where: is the original data; , are the maximum and minimum values in the original data respectively.

[0010] Both the training set and the test set contain 40 feature data and 1 label data; the 40 feature data contained in both the training set and the test set are respectively 20 feature data from meteorological data, 5 feature data from AIS data, 11 feature data from ship static data, 4 feature data from industry knowledge data, and 1 label data from AIS data; the 1 label data contained in the training set and the test set is the label data from AIS data; Among them, in the meteorological data, the 20 feature data from meteorological data are formed by further dividing 10 meteorological data into the meteorological data at the current moment and the meteorological data at a previous moment in the same voyage; In the AIS data, the speed at the current moment is set as the label data, and the speed at a previous moment in the same voyage is used as 1 feature data from AIS; meanwhile, the course of the ship is also divided into the course at the current moment and the course at a previous moment in the same voyage as 2 feature data from AIS; in addition, the latitude and longitude position data of the current ship are also used as 2 feature data from AIS.

[0011] The sample set is split using a random partitioning method to ensure that there are no duplicate samples in the training set and the test set. 90% of the samples are randomly selected from the pre - processed sample set as the data for the training set, and the remaining 10% of the samples are used as the data for the test set.

[0012] The learning model includes but is not limited to LightGBM, XGBoost, random forest, and BP neural network.

[0013] The learning model continuously optimizes the model parameters by minimizing the loss function until the optimal parameters are found.

[0014] The loss function to be minimized is: where: is the weight of the loss function based on data, is the weight of the loss function based on knowledge; and It can be various error measurement methods, including but not limited to Root Mean Square Error (RMSE), Mean Square Error (MSE), and Mean Absolute Error (MAE).

[0015] Adopting the above solution has the following advantages: 1. Since the present invention integrates meteorological data, AIS data, ship static data, and industry knowledge data, and adopts a "data + knowledge" hybrid-driven training mode to predict ship speed. This method effectively addresses the small sample problem in shipping operations by combining data and knowledge within the industry, significantly improving the accuracy of speed prediction and reducing the construction cost.

[0016] 2. In recent years, ship speed prediction methods based on artificial intelligence have mainly considered the ship's still water speed and the environmental impact factors on the ship as model inputs. The mapping relationship can be understood as "the still water speed at point A + the meteorological information at point A = the actual speed at point A". For the ship's still water speed, some dynamic variables that can reflect the set sailing speed of the ship are required, such as the main engine speed and the propeller shaft speed. The main engine speed and the propeller shaft speed are dynamic information during the ship's travel, and these parameters change with the ship's operating state. In real shipping operations, it is often the case that the model starts to execute the operation of predicting the speed at a certain location point, and the output is the speed for consecutive days in the future. During the process of model prediction, it is impossible to obtain the future ship dynamic information. Therefore, it is difficult and not suitable to use these artificial intelligence training schemes that combine dynamic information according to the real business needs of shipping companies. Based on the above problems, the present invention cleverly combines AIS real-time data information during the construction of the data set. Specifically, the AIS information at a certain previous moment (where "a certain" represents not fixed) is used as the feature for the subsequent moment. The mapping relationship is "the AIS information at a certain previous moment + the meteorological information corresponding to this point + the meteorological information at point A = the actual speed at point A". This construction scheme no longer requires the still water speed, can be applied to real weather routing services, and effectively extends the prediction duration, ensuring accurate prediction for a long time. Description of the Drawings

[0017] Figure 1 is a schematic flow chart of the data and knowledge hybrid-driven ship speed prediction method of the present invention; Figure 2 is the learning flow chart of the learning model in the data and knowledge hybrid-driven ship speed prediction method of the present invention; Figure 3 is a comparison chart of the Root Mean Square Error (RMSE) of the test set in a specific implementation case; Figure 4 is a result comparison diagram in a real business scenario; Figure 5 is a schematic diagram of Table 4 in a specific implementation case; Figure 6 It is the meaning table of factors in specific implementation cases; It is the schematic diagram of Table 5 in specific implementation cases; Figure 7 It is the schematic diagram of Table 6 in specific implementation cases; Figure 8 It is the schematic diagram of Table 7 in specific implementation cases; Figure 9 It is the schematic diagram of the model and error metric selection table of the experiment in specific implementation cases. Figure 10 Specific implementation manners The present invention will be further described in detail below with reference to the accompanying drawings.

[0018] As shown in

[0019] As Figure 1 shown, the data and knowledge hybrid-driven ship speed prediction method of the present invention includes the following steps: The first step is to collect the original data and preprocess the original data to obtain the standardized data .

[0020] The original data collected in this embodiment includes meteorological data, AIS data, ship static data, and industry knowledge data, which are specifically as follows: 1. Meteorological data Table 1 Meteorological characteristic data and data units

[0021] 2. AIS data Table 2 AIS data and data units

[0022] 3. Ship static data Table 3 Ship static data and data units

[0023] 4. Industry knowledge data The industry knowledge data is obtained by constructing an empirical formula through statistical analysis of a large amount of measured and experimental data.

[0024] The empirical formulas that can be adopted include but are not limited to the following methods: (1) Central Marine Research Institute of Russia (1) Wherein: is the actual speed of the ship in waves; is the speed of the ship in still water; is the wave height; is the angle between the ship's heading and the wave direction; is the actual displacement of the ship (unit: ton).

[0025] (2) Qingdao Meteorological Navigation Research Consortium (2) Wherein: is the actual speed of the ship in waves; is the significant wave height; is the angle between the ship's bow direction and the wave approaching direction; is the actual displacement of the ship; is the speed of the ship in still water; are the performance coefficients of the ship, wherein: = 0.745, = 0.05015, = 0.0045, = 1.35×10 -6 ; G is an empirical coefficient, and the specific value of G is shown in Table 4, that is, as Figure 5 shown: (3) National Meteorological Center, China Meteorological Administration ① Applicable to 20,000 - 50,000 deadweight tonnage cargo ships: (3) ② Applicable to 50,000 - 100,000 deadweight tonnage cargo ships: (4) Wherein: is the stall factor of the ship's navigation. Mark the wind direction (WD), wind speed (WS), wave height (SH), wave direction (SWD), swell height (SWH), displacement (TON), course (HDG) and speed (SPD), then the meanings of the above factors are as Figure 6 shown; The final speed formula is: (5) Wherein: is the actual speed of the ship in waves; is the speed of the ship in still water; is the stall factor of the ship's navigation.

[0026] (4) Semi - empirical formula (Kwon, Y. J. estimation method) (6) Wherein: is the actual speed of the ship in waves; is the speed of the ship in still water; is the stall direction factor varying with the wind direction and wind force level ( ), for details see Table 5, as shown in Figure 7 ; is the stall correction factor varying with the block coefficient, loading condition and Froude number ( ), for details see Table 6, as shown in Figure 8 ; is the ship type factor related to the ship type, wind force level ( ) and the ship displacement in terms of , for details see Table 7, as shown in ; Figure 9 ;

[0027] Among them: refers to the Beaufort wind scale, which is divided into 13 levels from 0 to 12 according to the wind strength.

[0028] Among them: That is, the calculation formula of the Froude number is: (7) Among them, is the speed of the ship in still water; is a fixed value, ; is the length of the ship's waterline.

[0029] In this embodiment, the process of data preprocessing is as follows: 1. Outlier handling In the ship dataset, the reasons for the occurrence of outliers include but are not limited to sensor failures, data transmission errors, and human recording errors. These outliers pose a potential threat to the performance of the model. Given the sufficient amount of data collected, the present invention is guided by data science and rationality and adopts the deletion method to deal with the outlier problem. Specifically, the following rules are set in this solution to identify and eliminate abnormal data: (1) If all the meteorological data in a certain piece of data shows 0, this data is eliminated; Specifically, if all the meteorological data in a certain piece of data shows 0, this usually means that the sensor may have failed, resulting in the inability to correctly collect and record meteorological information. Therefore, such data is regarded as abnormal data and is deleted.

[0030] (2) If the ship speed is lower than 4 knots or higher than 20 knots, this data is eliminated.

[0031] Specifically, generally speaking, when the ship speed is 3 knots, 2 knots or less, it is often a manual speed reduction operation when the ship is about to stop sailing. Therefore, if the ship speed is lower than 4 knots or higher than 20 knots, the present invention believes that this exceeds the normal sailing speed range. Therefore, these data are also regarded as abnormal and deleted.

[0032] (3) If the wind force level is less than 4, and the ship speed during normal sailing falls between 4 and 5 knots, this data is excluded.

[0033] Specifically, the Beaufort wind force level When it is less than 4, it means that the current is in a windless or light wind state. Under such weak wind force, if the ship speed during normal sailing falls between 4 and 5 knots, the present invention believes that this is an unreasonable phenomenon. Therefore, such data is also regarded as abnormal and deleted.

[0034] (4) If the ship charter speed is 0, this data is excluded.

[0035] Specifically, the ship charter speed refers to the sailing speed agreed in the ship charter contract. If the ship charter speed in a certain piece of data is 0, this usually means that there is an abnormality in the data record. Therefore, the present invention determines such data as abnormal and deletes it.

[0036] (5) If the wind force level is equal to 0, the data with the difference between the ship speed and the charter speed greater than 8 is excluded.

[0037] Specifically, when is equal to 0 (that is, it means that the current is in a windless state), the present invention regards the data with the difference between the ship speed and the charter speed greater than 8 as abnormal data and deletes it.

[0038] 2. Encoding processing The ship loading state is converted using 0-1 encoding: Among them, represents empty load, represents full load, that is, if the ship is empty loaded, the code is 0; if the ship is full loaded, the code is 1, that is: if the ship is empty loaded, the code is 0; if the ship is full loaded, the code is 1. In this way, the input data is properly formatted into a numerical type to meet the requirements of model training.

[0039] 3. Normalization processing In order to eliminate the differences in dimension and numerical range between features, the present invention adopts the Min-Max normalization method, which converts the data into the range of [0,1] through the following formula: (9) wherein: is the standardized data; is the original data; and are respectively the maximum and minimum values in the original data.

[0040] In summary, this processing not only accelerates the process of the gradient descent algorithm to find the optimal solution, but also effectively improves the prediction accuracy of the model; In the second step, all data are divided into multiple training sets and test sets. Both the training set and the test set contain feature data for inputting into the learning model and label data for verifying the prediction results of the learning model.

[0041] Both the training set and the test set contain 40 feature data and 1 label data. The 40 feature data contained in both the training set and the test set are respectively 20 feature data from meteorological data, 5 feature data from AIS data, 11 feature data from ship static data, 4 feature data from industry knowledge data, and 1 label data from AIS data. The 1 label data contained in both the training set and the test set is the label data from AIS data; In meteorological data, the 20 feature data from meteorological data are formed by further dividing 10 meteorological data into the meteorological data at the current moment and the meteorological data at a previous moment in the same voyage.

[0042] Specifically, the 10 meteorological data are further divided into the meteorological data at the current moment and the meteorological data at a previous moment (such as moment A) in the same voyage. Therefore, at the input end of the model, a total of 20 meteorology-related data features are used.

[0043] In AIS data, the speed at the current moment is set as the said label data, while the speed at a previous moment in the same voyage is used as 1 feature data from AIS; meanwhile, the course of the ship is also divided into the course at the current moment and the course at a previous moment in the same voyage as 2 feature data from AIS; in addition, the latitude and longitude position data of the current ship are also used as 2 feature data from AIS.

[0044] In AIS data, the speed at the current moment is set as the said label data, while the speed at a previous moment in the same voyage is used as 1 feature data from AIS; meanwhile, the course of the ship is also divided into the course at the current moment and the course at a previous moment in the same voyage as 2 feature data from AIS; in addition, the latitude and longitude position data of the current ship are also used as 2 feature data from AIS.

[0045] Specifically, in the AIS data, the ship speed at the current moment is set as the labeled data, while the ship speed at a previous moment (e.g., moment A) during the same voyage is used as one of the feature data. Similarly, the ship's heading is also divided into the heading at the current moment and the heading at a previous moment (such as moment A) during the same voyage, both of which are used as feature data. In addition, the latitude and longitude position data of the current ship are also included in the scope of feature data.

[0046] Specifically, the training set is specifically used for the training process of the model, while the test set is used to verify and evaluate the performance of the model. In this solution, the sample set is split using a random partitioning method to ensure that there are no duplicate samples in the training set and the test set. 90% of the samples are randomly selected from the preprocessed sample set as the training set, and the remaining 10% of the samples are used as the test set. At the same time, to ensure the effectiveness of the model, the samples in the training set and the test set need to maintain a high degree of correlation to accurately reflect the performance of the model in the actual application scenario.

[0047] In the third step, the training set and the test set are input into the learning model to train the learning model until the optimal parameters are found and the optimal parameters are saved.

[0048] Specifically, as Figure 2 shown, first, the training set and the test set are input into the learning model. The training set and the test set together form the basis for training the machine learning model. The model includes but is not limited to LightGBM, XGBoost, random forest, and BP neural network. In the training stage, the model parameters are continuously optimized by minimizing the loss function until the optimal parameters are found, and these optimal parameters are saved. The test set is used to verify the optimal parameters to ensure the correctness of the optimal parameters.

[0049] In the fourth step, the new test set containing historical data and industry knowledge data is input into the prediction model to achieve the prediction of the ship speed.

[0050] For the specific implementation case, aiming at the small-sample problem in the shipping business, this solution proposes an innovative solution.

[0051] Based on this solution, the combination in Table 8 is selected for experiments. Table 8 is as Figure 10 shown.

[0052] The data used in the experiments of the present invention cover eight ship types such as dry bulk carriers, open hatch cargo ships, general cargo ships, and ore carriers. The experimental results are as follows.

[0053] As Figure 3As shown, 100 voyages of data were selected for test verification in the experiment. The test results show that compared with the root mean square error (RMSE) of 2.313 obtained by the empirical statistical method, the "data + knowledge" dual - drive training scheme proposed by the present invention significantly reduces the RMSE to 1.009, and the overall error is reduced by up to 56.37%.

[0054] The innovative scheme of the present invention has been applied in actual shipping operations and has won good feedback due to its high accuracy.

[0055] As Figure 4 shown, on the left is the planned route from the port of Lirquén to the port of Singapore, passing through the Cape of Good Hope area with severe weather conditions. On the right, the test results of the present invention's scheme and the empirical statistical method are compared when passing through this Cape of Good Hope area on August 7, 2024. In the figure, the horizontal axis represents time, specifically 0:00, 6:00, 12:00, and 18:00 on August 7, 2024; the vertical axis is the absolute value of the difference between the predicted ship speed and the actual ship speed recorded by AIS (Automatic Identification System) at each time point for the two methods. Obviously, the speed error obtained by the present invention's scheme is significantly lower than that of the empirical statistical method, indicating that its predicted value is closer to the actual ship speed. In particular, at 12:00 on August 7, the gap between the predicted value and the actual value is reduced to only 0.06 knots.

[0056] In summary, after shipping companies adopt the model trained by the present invention's scheme, they can accurately deduce and plan the safest route to avoid severe weather, which not only effectively reduces economic losses but also ensures the safety of crew members.

[0057] This scheme innovatively adopts a "data + knowledge" hybrid - drive training mode, integrating meteorological data, AIS data, ship static data, and industry knowledge data, significantly improving the accuracy of ship speed prediction, effectively addressing the data scarcity problem in shipping operations, and promoting the process of shipping intelligence.

Claims

1. A method for predicting ship speed driven by a hybrid of data and knowledge, characterized in that, The steps include: First step, collect the original data and preprocess the original data to obtain the standardized data ; among them, the original data includes meteorological data, AIS data, ship static data and industry knowledge data; Step 2: All data are divided into multiple training sets and test sets. Both the training sets and the test sets contain feature data for inputting into the learning model and label data for verifying the prediction results of the learning model; The third step is to input the training set and the test set into the learning model to train the learning model until the optimal parameters are found and the optimal parameters are saved; In the fourth step, the new test set containing historical data and industry knowledge data is input into the prediction model to predict the ship speed.

2. The data and knowledge hybrid-driven ship speed prediction method according to claim 1, characterized in that The industry knowledge data is obtained by calculating the empirical formula; the empirical formula includes the formula of the Russian Central Institute of Shipping, the formula of Qingdao Meteorological and Navigation Research Consortium, the formula of the National Meteorological Center of the State Meteorological Administration and / or a semi-empirical formula; wherein the semi-empirical formula is: , Among them, is the actual speed of the ship in waves; is the speed of the ship in still water; is the stall direction factor varying with the wind and wave direction and wind force level ; is the stall correction factor varying with the block coefficient, loading condition and Froude number ; is the ship form factor related to the ship form, wind force level and the ship displacement in terms of ; ​ 3. The data and knowledge hybrid-driven ship speed prediction method according to claim 1, characterized in that In the first step, the process of preprocessing the original data is as follows: S1 removes abnormal data. The specific rules are: (1) If all meteorological data in a certain data are displayed as 0, the data will be eliminated; (2) If the speed is less than 4 knots or greater than 20 knots, the data shall be discarded; (3) Wind force level If it is less than 4, and the ship speed during normal navigation falls between 4 and 5 knots, this data shall be excluded; (4) If the ship charter speed is 0, the data will be eliminated; (5) Wind force level When it is equal to 0, the data with a difference greater than 8 between the ship's speed and the charter speed shall be excluded; S2 encoding process: Use 0-1 coding to convert the ship loading status: , Among them, is no-load, is full-load, that is, if the ship is no-load, the code is 0; if the ship is full-load, the code is 1; S3 normalization processing: The normalization method is used to standardize the data and convert the data into the data within the range of [0, 1]. .

4. The data and knowledge hybrid-driven ship speed prediction method according to claim 3, wherein The normalization is a Min-Max normalization method, which converts data to the interval [0, 1] by the following formula: , Among them, is the original data; and are the maximum value and the minimum value in the original data respectively.

5. The data and knowledge hybrid-driven ship speed prediction method according to claim 1, wherein The training set and the test set both contain 40 feature data and 1 label data; the 40 feature data contained in the training set and the test set are 20 feature data from meteorological data, 5 feature data from AIS data, 11 feature data from ship static data, 4 feature data from industry knowledge data and 1 label data from AIS data; the 1 label data contained in the training set and the test set is the label data from AIS data; Among them, in the meteorological data, 20 characteristic data from the meteorological data are formed by 10 meteorological data being further subdivided into meteorological data at the current moment and meteorological data at a previous moment in the same voyage; In the AIS data, the current speed is set as the tag data, and the speed at a previous time in the same voyage is used as one feature data from AIS; at the same time, the ship's heading is also subdivided into the current heading and the heading at a previous time in the same voyage as two feature data from AIS; in addition, the current ship's latitude and longitude position data are also used as two feature data from AIS.

6. The data and knowledge hybrid-driven ship speed prediction method according to claim 1, wherein The sample set is split randomly to ensure that there are no duplicate samples in the training set and the test set. 90% of the samples are randomly selected from the preprocessed sample set as the data of the training set, and the remaining 10% of the samples are used as the data of the test set.

7. The data and knowledge hybrid-driven ship speed prediction method according to claim 1, wherein The learning models include but are not limited to LightGBM, XGBoost, random forest and BP neural network.

8. The data and knowledge hybrid-driven ship speed prediction method according to claim 1, characterized in that The learning model continuously optimizes the model parameters by minimizing the loss function until the optimal parameters are found.

9. The data and knowledge hybrid-driven ship speed prediction method according to claim 8, wherein The minimization loss function is: , Wherein: is the weight of the data-based loss function, i.e., the weight of the knowledge-based loss function; and can be various error measurement methods, including but not limited to root mean square error RMSE, mean square error MSE, and mean absolute error MAE.

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