Research on the Bearing Lifespan Prediction Method for Ship Propulsion Shaft Systems Based on an Enhanced Domain Adversarial Neural Network

Abstract

To address the challenge of accurate lifespan prediction for bearings in different operating conditions within ship propulsion shaft systems, a two-stage prediction model based on an enhanced domain adversarial neural network (DANN) is proposed. Firstly, pre-training features containing comprehensive degradation information are extracted from the entire source domain dataset encompassing all operational conditions. Subsequently, DANN is employed to extract domain-invariant features that are difficult to distinguish. Following this, a feature alignment process is utilized to align high-dimensional features with pre-training features, thereby mitigating the adverse effects caused by missing data in the incomplete target operational condition dataset. Finally, the effectiveness of this approach is validated using operational data from bearings under multiple operating conditions. The experimental results demonstrate that the method presented in this paper achieves an average error reduction of 0.0626 and 0.0845 compared to the MK-MMD transfer learning method and self-attention ConvLSTM algorithms, respectively, and exhibits higher predictive reliability. This method can provide valuable insights for lifespan prediction challenges concerning bearings in ship propulsion shaft systems under various operational conditions, as well as similar cross-domain lifespan prediction problems.

1. Introduction

Rolling bearings are intricately integrated into the framework of shipbuilding and manufacturing technologies, performing a vital role in a myriad of applications, from marine steam turbines and aft power compartments to connecting pipelines and propulsion shaft systems. Within the ship propulsion system, the bearings endure a variety of stressors including vibration [1], torsion [2], and coupled axial [3] and lateral [4] loads. A failure of these bearings can precipitate the breakdown and malfunction of system components, and, in extreme cases, the structural integrity of the entire ship, potentially leading to catastrophic outcomes [5,6].

In the literature [7], the remaining useful life (RUL) is defined as the remaining time for which a device or system can operate according to its primary task. This concept has been widely applied in the field of equipment health management. Currently, research on RUL prediction mainly focuses on two methods: physics-based models and data-driven methods. Due to the elimination of the need for extensive expert knowledge and the development of complex physical models in data-driven methods, they have attracted considerable attention from scholars both domestically and internationally.

Traditional data-driven methods include support vector machines, Bayesian methods, and BP neural networks [8,9,10]. The literature [11] integrates methods such as principal empirical mode decomposition (PEEMD), principal component analysis (PCA), and support vector machines (SVM), applying them to the diagnosis of faults in ship propulsion shaft system rolling bearings and tail bearings. With the advent of big data and advances in computer technology, deep neural networks exemplified by convolutional neural networks (CNN) have gained widespread attention in the field of life prediction. Li et al. [12] applied neural networks to the scenario of predicting the remaining life of aero-engines, utilizing the neural network to automatically extract life features and predict life. Kang et al. [13] investigated the long short-term memory (LSTM) model used for the life prediction of rocker gearboxes. Wan Shaoke et al. [14] extracted local spatial feature information through CNNs, extracted time information through LSTM, and used ConvLSTM to predict the RUL of azimuth, thus improving prediction accuracy. Subsequently, Ren et al. [15] added an attention mechanism to the ConvGRU model, improving prediction efficiency while ensuring accuracy.

In the field related to ship propulsion shaft system bearings, studies in the literature [16,17,18] have used back propagation neural networks (BPNNs) and symmetrical dot pattern (SDP) visual diagnostic methods to diagnose the health of the ship propulsion shaft system and achieved satisfactory results.

While deep learning-based lifetime prediction methods have shown promising results, they are based on the premise that both the training and prediction datasets come from similar operating conditions and span the entire lifecycle of data with health labels. These methods can only diagnose the health of bearings operating under the same conditions and cannot predict targets operating under different conditions. This is because different operating conditions and sensor placement schemes may lead to different data distributions, resulting in different data features. The differences between data features hinder the generalization of learning knowledge from training data to test data. As shown in Figure 1, the difference in data distribution between the source domain and the target domain leads to the initial degradation and termination position of the source domain data before the target domain data, which is represented as a domain gap. When training on source domain data, the model can accurately predict the life labels within the same domain. However, when the same model is applied to target domain data characterized by different data distributions, inaccurate predictions may occur. This phenomenon is known as domain shift (DS), which is discussed in reference [19].

As stated in references [20,21,22], due to the effectiveness of transfer learning and domain adaptation techniques in solving the domain shift problem, they have gradually been incorporated into the research of remaining useful life (RUL) prediction. The literature [23,24,25] suggests using metrics such as maximum mean discrepancy (MMD) and correlation alignment (CORAL) to mitigate the distribution difference between source and target domain data, thereby extracting domain-invariant features. The concept of adversarial learning introduced in 2014 provides a new approach to domain adaptation objectives. Ganin et al. [26] proposed the domain adversarial neural network (DANN), which includes a gradient reversal layer (GRL). This design can extract training and source domain data that the domain discriminator cannot distinguish, thus promoting effective domain transfer. Subsequent improvements have been made to this model; Wang Y [27] adopted an alternating maximum–minimum optimization strategy to train generative adversarial networks (GANs) for data feature extraction. Li et al. [28] advocate applying DANN to data comparisons in different regions. They used generative adversarial networks (GANs) to identify the data distribution under optimal device health conditions, thereby enhancing the ability to transfer knowledge across different domains.

While the transfer learning bearing prediction algorithm based on DANN has demonstrated the ability to predict the health condition of bearings in various operating scenarios using data from a single condition, a significant limitation is that it relies on comprehensive lifecycle data from the target bearing. Given the complexity of shipboard equipment and the constraints of onboard space, data collection has proven to be challenging.

Therefore, typically, only a limited set of early bearing degradation data is available. To address this challenge, this paper introduces an improved DANN method for predicting the remaining useful life (RUL) of bearings under conditions of limited target-bearing data. The structure of the remainder of this paper is as follows: in Section 2, we delve into the basic principles of the ship propulsion shaft system and the DANN model. Section 3 elaborates on our proposed two-layer prediction method, aimed at enhancing the DANN model. The effectiveness of this method is confirmed through the analysis of bearing data under various operating conditions, as shown in Section 4. A comprehensive summary and the conclusion of the research are presented in Section 5.

2. Methodology Overview

2.1. Problem Description

As a pivotal element of a ship’s propulsion mechanism, the propulsion shaft system is tasked with the critical role of conveying the main engine’s kinetic energy to the propeller, thereby facilitating the movement of the vessel. This complex assembly is composed of an array of integral components, including but not limited to the propeller, stern tube, intermediate shaft, gearbox, and main engine. Embedded within this intricate configuration, rolling bearings are dispersed at strategic junctures, underscoring their prominence as key components within the system. Figure 2 illustrates a prototypical propulsion shaft system, encompassing elements such as a motor, gearbox, coupling, bearings, and a propeller (load). Within this setup, the rolling bearings undertake the transmission function. Bearings, subject to varying radial forces and rotational speeds, manifest diverse wear patterns and operational health conditions. Under these circumstances, the precision in forecasting the remaining useful life (RUL) of the bearings, capable of adapting to assorted operational scenarios, emerges as the fundamental building block in the optimization of the health management for the propulsion shaft system.

Due to the confined space within the ship’s engine room and the harsh operating environment during navigation, data collection for bearings is challenging. Therefore, this study explores the remaining useful life (RUL) prediction for target bearings based on limited vibration data, comprehensive experimental lifecycle data of the bearing, and transfer learning domain adaptation methods. The complete cycle-bearing data collected from the experimental platform are designated as the source domain data, denoted as $D_s={(x_i^s,y_i^s)}_{i=1}^{n_s}, x_i^s\in X_s,y_i^s\in Y_s$. The incomplete bearing data to be predicted for its lifecycle are referred to as the target domain data, denoted as $D_t={(x_i^t)}_{i=1}^{n_t}, x_i^t\in X_t$. Here, $x_i^s$ is the i-th sample in the source domain; $x_i^t$ is the i-th sample in the target domain; $X_s$ is the set of all samples from the source domain; $X_t$ is the set of all samples from the target domain; $y_i^s$ is the label for the i-th sample; $Y_s$ is the set of all distinct labels; $n_s$ is the total number of samples in the source domain; and $n_t$ is the total number of samples in the target domain. Because the operating conditions of the bearings differ, $D_s$ and $D_t$ have different marginal distributions. It is worth noting that the source domain data are the run-to-failure data with life labels, while the target domain data are label-free and incomplete.

In most cross-domain prediction problems, the common practice is to align the features of the source and target domains and map them to a new feature space. However, due to the large amount of missing data in the target domain in this problem, it will affect the alignment of features, leading to inaccurate predictions. Therefore, this paper aims to extract the common degradation information of the two domains under the condition of incomplete target domain data to achieve cross-condition life prediction.

2.2. Introduction to DANN Model

The DANN (domain adversarial neural network) model, proposed by Ganin [26], is a transfer learning network based on the adversarial learning approach. In recent years, it has been widely applied to address the issue of cross-condition prediction in bearing systems. The experimental bearing data, consisting of complete operational data and labeled data, are defined as the source domain data, while the data from other operational conditions for remaining life prediction are defined as the target domain data. The fundamental idea of the model involves introducing a domain discriminator into both the feature extractor and the predictor. By updating the parameters of the domain discriminator, it is endowed with the ability to discriminate between the source and target domains. Concurrently, the feature extractor parameters are optimized to extract features that are difficult to distinguish, thereby reducing the distribution differences among different operational conditions.

In terms of its specific structure, the DANN model consists of three components: the feature extractor $M_F$, the domain discriminator $M_D$, and the predictor $M_P$. To implement the adversarial strategy between the domain discriminator $M_D$ and the feature extractor $M_F$, a gradient reversal layer (GRL) is added between them, as detailed in the literature [26]. The DANN loss function includes two components, $L_P$ and $L_D$, defined by Equation (1) and Equation (2), respectively:

$$L_P={\displaystyle \sum _{i=1}^N{L}_p^i(p,y_i)}$$

(1)

$$L_D={\displaystyle \sum _{i=1}^N{L}_d^i(d,d_i)}$$

(2)

where $L_P^i$ represents the regression loss value for the i-th predicted value $p$, denotes the prediction from the predictor, and $y_i$ denotes the label value for the i-th sample’s remaining life. $L_P^i$ represents the domain discrimination loss value for the i-th sample, $d$ denotes the output of the domain discriminator, and $d_i$ represents the domain label for the i-th sample. When the features come from the source domain, $d_i$ is set to 1; otherwise, it is set to 0. The overall loss function for the DANN model can be defined using the following Equation (3):

$$f_{DANN}=L_P-\lambda L_D$$

(3)

where $\lambda$ is the weighting factor that balances the importance of the two loss components.

3. The Enhanced DANN Two-Stage Prediction Method

Due to the complexity of the operation conditions of various types of mechanical equipment on ships and the difficulty of sensor installation and data collection, it is challenging to obtain bearing data covering the entire range from healthy operation to failure. Only a small amount of bearing vibration data can be collected in the early degradation stage. Therefore, this paper proposes a two-stage prediction model based on improved cross-entropy DANN. The model consists of two stages. A deep learning network constructed upon source domain data is employed in the first stage. This network learns the relationship between the source domain data and life expectancy labels, extracting features encompassing the progression from a healthy state to severe degradation. Features derived during this phase offer regulative references for the feature extraction of the subsequent DANN stage. In the second phase, the model initially capitalizes on the adversarial domain adaptation concept to optimize the domain discriminator and feature extractor parameters, ensuring that the extracted features exhibit both domain discriminative and invariant characteristics. Concurrently, to mitigate the adverse effects induced by the incomplete data in the target domain, a regularization term is incorporated. This term ensures alignment between the features extracted in this stage and the pre-trained features. As a result, while retaining domain discriminative and invariant properties, the extracted features also encapsulate comprehensive cycle information from a healthy state to severe degradation, thereby diminishing the detrimental impact caused by missing data in the target domain.

3.1. Feature Pre-Extraction Stage

In this stage, a deep learning regression model is developed, which consists of a feature extractor and a predictor. The training data include experimental bearing data with complete vibration cycle data and remaining life labels to ensure that the extracted features capture the complete degradation pattern. The extracted features can serve as a reference for feature alignment in the second stage. In this stage, the feature extractor is denoted as $M_F^{\ast }$, the predictor is denoted as $M_P^{\ast }$, and the predicted result of the predictor is denoted as $p^{\ast }$. The regression loss in the first stage, $L_p^{\ast }$, is defined by Equation (4):

$$L_p^{\ast }=\frac{{\displaystyle \sum {}_{i=1}^n\left|p^{\ast }-y_i\right|}}{n}$$

(4)

In the context of the provided Equation (4), it can be inferred that $n$ represents the number of features. The given loss function is indeed the only loss function used in this stage.

3.2. Adaptation Stage of the Adversarial Domain

In this stage, based on the DANN model, the degradation information contained in the source domain data is transferred to the target domain data by combining the features extracted in the first stage. Traditional DANN models typically involve two loss functions, which may lead to the extraction of indistinguishable features after multiple alignment iterations. Consequently, the correlation between the extracted features and the degree of equipment degradation may decrease. To overcome this limitation, this paper proposes the introduction of cross-entropy (CE) as a metric to realign the features, taking into account the pre-training features that contain information throughout the equipment’s lifecycle, which were extracted in the first stage.

Specifically, during the training phase of DANN, in addition to the regression loss $L_p$ and the domain discrimination loss $L_D$, a cross-entropy loss $L_{cm}$ is introduced to measure the alignment between the extracted features and the pre-training features. The cross-entropy loss $L_{cm}$ is defined by Equation (5):

$$L_{cm}=H(f_i^{\ast }(x_i),f_i(x_i))={\displaystyle \sum _{i=1}^n{f}_i^{\ast }(x_i)\log \frac{1}{f_i(x_i)}}$$

(5)

Indeed, in the aforementioned equation, $f_i^{\ast }(x_i)$ and $f_i(x_i)$ represent the i-th feature of the pre-trained model and the DANN model, respectively. $H$ denotes the calculation process for obtaining cross-entropy.

Taking into account the original loss function of DANN, the overall optimization loss for the second stage can be defined as Equation (6):

$$L_{All}=L_p+{\lambda }_1{L}_{cm}+{\lambda }_2{L}_D$$

(6)

The penalties of consistency regularization ${\lambda }_1$ and domain discrimination ${\lambda }_2$ are defined in this study. The framework and structure diagram of this method are illustrated in Figure 3.

4. Experimental Study

4.1. Introduction to the Experimental Platform

To affirm the efficacy of the proposed method for predicting bearing life across varying operational conditions, this study employs the bearing dataset supplied by the FEMTO-ST Institute for the PHM2012 Data Challenge [29]. The PRONOSTIA experimental platform is capable of expediting bearing degradation under either constant or fluctuating operational conditions while concurrently gathering real-time health monitoring data, including rotation speed, load force, temperature, and vibration. Crucially, the structure of this experimental platform mirrors that of a ship propulsion shaft system. Thus, the diverse operational condition bearing data procured from the PRONOSTIA platform can substantiate the validity of the method proposed herein for predicting bearing performance across different operational conditions within a ship propulsion shaft system.

4.1.1. Outline

PRONOSTIA, depicted in Figure 4, serves as a specialized experimental platform intended for the testing and validation of bearing fault detection, diagnosis, and prognostic methodologies. The paramount goal of PRONOSTIA is to furnish genuine experimental data, illustrating the degradation trajectory of ball bearings over their entire lifespan, up to the point of total failure. PRONOSTIA is fundamentally composed of three sections: the rotating segment, the degradation induction segment (where radial force is exerted on the bearing under test), and the measurement segment. Detailed descriptions are provided subsequently.

4.1.2. Rotating Part

This part includes the asynchronous motor with a gearbox and its two shafts: the first one is near to the motor and the second one is placed at the ride side of the incremental encoder. The motor has a power equal to 250 W and transmits the rotating motion through a gearbox, which allows the motor to reach its rated speed of 2830 rpm, so that it can deliver its rated torque while maintaining the speed of the secondary shaft to a speed of less than 2000 rpm. Compliant and rigid shaft couplings are used to create connections for the transmission of the rotating motion produced by the motor to the shaft support bearing.

The bearing support shaft in Figure 5 leads the bearing through its inner race. This one is kept fixed to the shaft with a shoulder on the right hand and a threaded locking ring on the left hand. The shaft which is made of one piece is held by two pillow blocks and their large gearings. The two clampings allow the longitudinal blocking of the shaft between the two pillow blocks. A human machine interface allows the operator to set the speed, to select the direction of the motor’s rotation, and to set the monitoring parameters such as the motor’s instantaneous temperature expressed in percentage of the maximum temperature of use.

4.1.3. Loading Part

Components from this part are grouped in a unique and same aluminum plate partially isolated from the instrumentation part by a thin layer of polymer. The aluminum plate supports a pneumatic jack, a vertical axis and its lever arm, a force sensor, a clamping ring of the test bearing, a support test bearing shaft, two pillow blocks, and their large, oversized bearings. The force issued from the pneumatic jack is first amplified by a lever arm and is then indirectly applied on the external ring of the test ball bearing through its clamping ring, The details of the loading part are shown in Figure 6.

This loading part constitutes the heart of the global system. In fact, the radial force reduces the bearing’s life duration by setting its value up to the bearing’s maximum dynamic load, which is 4000 N. This load is generated by a force actuator, which consists of a pneumatic jack, where the supply pressure is delivered by a digital electro-pneumatic regulator.

4.1.4. Measurement Part

The operating conditions are determined by instantaneous measurement of the radial force applied to the bearing, the rotational speed of the shaft handling the bearing, and the torque applied to the bearing. Each of these three analog measurements was taken at a frequency equal to 100 Hz. The characterization of bearing degradation is based on vibration signals. The model of the vibration sensor is composed of two micro accelerometers located at 90° to each other; the first one is placed on the vertical axis, and the second one is placed on the horizontal axis. Two accelerometers are placed radially on the outer race of the bearing, and the acceleration measurement is sampled at 25.6 kHz.

4.1.5. Organization of Data

The experiment conducted by the FEMTO-ST Research Institute considered data representing three different loads:

-

First operating conditions: 1800 rpm and 4000 N;

-

Second operating conditions: 1650 rpm and 4200 N;

-

Third operating conditions: 1500 rpm and 5000 N.

The data collected in three different scenarios can simulate bearing data under three different operating conditions in the ship propulsion system. Detailed information about the bearing dataset is presented in

Table 1. Bearing dataset description.

Condition	Rotating Speed (rmp)	Radical Force (N)	Bearing Data
Condition 1	1800	4000	bearing1-1–bearing1-7
Condition 2	1650	4200	bearing2-1–bearing2-7
Condition 3	1500	5000	bearing3-1–bearing3-3

.

4.2. Data Preprocessing and Implementation Details

Three distinct data types are encompassed within the bearing dataset: vertical vibration, horizontal vibration, and temperature. Given the enhanced capability of the horizontal-axis accelerometer to discern more comprehensive degradation patterns, data from the other two categories were excised. To mitigate potential perturbations stemming from noise and anomalous values, a pre-processing step was undertaken using the max–min standardization method prior to model input, effectively constraining data to the [0, 1] interval. The governing equation for this max–min standardization is elucidated in Equation (7).

$${\overline{X}}_t=\frac{X_t-X_{\min }}{X_{\max }-X_{\min }}$$

(7)

Let $X_t$ denote the dataset value at time $t$, $X_{\min }$ be the smallest value within the dataset, and $X_{\max }$ stand as its maximal counterpart. Following normalization, the value at time $t$ is represented by $R^2$.

Within the model, the feature extractor is composed of three convolutional layers, each endowed with 32, 16, and 1 kernel(s), accompanied by three filters. Subsequent to the terminal convolutional layer, a flattening layer seamlessly integrates into a fully connected layer.

Both the RUL predictor and the domain discriminator modules predominantly consist of fully connected layers. The sophisticated features procured by the feature extractor act as inputs for these modules. The predictor yields an output characterized by a single neuron, indicative of the RUL value, while the discriminator offers a dual-neuron output, each representing separate domains. A comprehensive visualization of each module’s network structure is showcased in Figure 7, with diverse parameter specifications and their corresponding values cataloged in

Table 2. The parameters used in this method.

Parameters From	Parameters	Value
Overall	Learning rate	1 × 10−5
	Batch size	64
	Epochs	4000
Feature Extractor	Conv1	32 × 3
	Conv2	16 × 3
	Conv3	1 × 3
	Neurons Number in FC1	30
Predictor	Neurons Number in FC2	10
	Output of the RUL predictor	1
Domain Discriminator	Neurons Number in FC3	100
	Output of the Domain Discriminator	2

. Notably, throughout the network, Relu serves as the predominant activation function, with the exception of the concluding layer that leverages Softmax. During the model’s training phase, the integration of dropout techniques stands pivotal in augmenting its efficacy.

This section is divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

4.3. Life Prediction Process

To demonstrate the effectiveness of cross-condition prediction, this experiment employed a mutual validation approach among three different operating conditions. Due to the limited amount of data for operating condition 3, it was not used as part of the training set. The experiment included three groups: predicting operating condition 1 based on operating condition 2, predicting operating condition 2 based on operating condition 1, and predicting operating condition 3 based on operating condition 1. The specific dataset division is presented in

Table 3. Dataset partitioning.

Task	Source Data (Labeled)	Target Data (Unlabeled)	Notes
Task 1	bearing2-1–bearing2-7	bearing1-3	Condition2→ Condition 1
Task 2	bearing1-1–bearing1-7	bearing2-3	Condition 1→ Condition 2
Task 3	bearing1-1–bearing1-7	bearing3-3	Condition 1→ Condition 3

:

To verify the effectiveness of the improvement strategy of this method, the experimental effects of this method and the unimproved DANN model were compared, and experiments were conducted on three groups of target bearings. The comparison of the three groups of experimental results is shown in Figure 8, and the specific experimental results are shown in

Table 4. Index values before and after improvement.

Task	Method	R2	MAE	MSE	RMSE
Task 1	DANN	0.3450	0.2831	0.0067	0.3262
	Improved DANN	0.9385	0.0572	0.0015	0.0732
Task 2	DANN	0.2273	0.2447	0.0066	0.2939
	Improved DANN	0.9035	0.0672	0.0020	0.0898
Task 3	DANN	0.1313	0.4625	0.0244	0.5078
	Improved DANN	0.8323	0.0913	0.0057	0.1182

. From the figure, it can be concluded that the prediction effect of the DANN two-stage model improved by cross-entropy is more accurate than the traditional DANN prediction result. This paper adopts four indicators to measure the prediction accuracy, namely the coefficient of determination ($R^2$), the mean absolute error (MAE), the mean square error (MSE), and the root mean square error (RMSE). From the table, it can be seen that the various indicators predicted by the improved model are significantly better than before the improvement. Among them, the absolute error and mean square error, two error indicators, decreased by 77.56% and 70.66%, respectively, after the transfer; the coefficient of determination increased on average by 0.66 after applying the transfer learning strategy, so it is necessary to improve the DANN model strategy.

4.4. Comparative Experiments

4.4.1. Introduction to Comparative Algorithms

In order to further verify the effectiveness of the improved DANN transfer learning method proposed in this study, the method proposed in this paper was compared with two other prediction algorithms, namely the MK-MMD [30]-based transfer learning method and the self-attention ConvLSTM (SA-ConvLSTM) [31] deep learning prediction method. Both methods have been proven to perform well in predicting cross-operating condition bearing problems.

MK-MMD: The MK-MMD transfer learning model proposed in the literature [30] uses the MK-MMD with multiple kernels to measure the differences between data distributions across domains. This model consists of a CNN feature extractor and an RUL predictor module, with its loss function encompassing regression loss and maximum mean discrepancy loss among others. This method is often used to solve cross-domain prediction problems.

SA-ConvLSTM: The SA-ConvLSTM used in the literature [31] is a deep neural prediction network. The model replaces the fully connected layers within the network structure with convolution operators to reduce network redundancy and enhance its non-linear modeling capabilities. It also designs an SA module and embeds it into the model. This method is superior to other traditional prediction methods in terms of convergence speed and prediction accuracy.

4.4.2. Comparative Experiment Process and Result Analysis

Initially, the data undergo preprocessing and standardization in accordance with the methodology delineated in Section 4.2. Following this, the MK-MMD transfer learning approach proposed in reference [30], the Self-Attention ConvLSTM (SA ConvLSTM) deep learning predictive technique put forward in reference [31], and the enhanced DANN algorithm introduced in this study are each employed for three cross-operating condition prediction tasks, utilizing identical source and target domain datasets as outlined in Table 3. This process yields a total of nine prediction results, the specifics of which are presented in

Table 5. Comparative experimental results of adaptation algorithms in different domains.

Task	Method	R2	MAE	MSE	RMSE
Task 1	MK-MMD	0.5855	0.1665	0.0041	0.1992
	SA-ConvLSTM	0.7817	0.1208	0.0029	0.1410
	Proposed	0.9385	0.0572	0.0015	0.0732
Task 2	MK-MMD	0.5631	0.1473	0.0044	0.1940
	SA-ConvLSTM	0.3400	0.1806	0.0055	0.2438
	Proposed	0.9035	0.0672	0.0020	0.0898
Task 3	MK-MMD	0.4194	0.2497	0.0138	0.2876
	SA-ConvLSTM	0.6183	0.1639	0.0097	0.2011
	Proposed	0.8323	0.0913	0.0057	0.1182

, while the graphical representations of these predictions are displayed in Figure 9.

As can be seen from Figure 9, the improved DANN prediction algorithm is superior to the other two methods. According to Table 5, the average coefficient of determination (R²) of the improved DANN prediction algorithm is 0.3114 and 0.3688 higher than that of the MK-MMD model and SA-ConvLSTM prediction algorithm, respectively. In addition, the average value of the three errors of the improved DANN prediction algorithm is 0.0562, which is 0.0626 and 0.0845 lower than the errors of the MK-MMD model and SA-ConvLSTM prediction algorithm, respectively. These results indicate that compared with the other two common transfer learning and prediction algorithms, the method proposed in this paper shows better accuracy and reliability in bearing cross-state prediction.

5. Conclusions

To further enhance the health diagnostic level of ship propulsion systems and address the difficulty in predicting bearing life in ship propulsion shaft systems due to the challenges in collecting full-cycle bearing data, this study proposes a remaining useful life (RUL) prediction method based on a cross-entropy improved domain adversarial neural network (DANN). The advantage of this method is that it only requires the vibration data of one complete cycle of a certain model of bearing and a small amount of target bearing data to be predicted to achieve a relatively accurate cross-condition prediction effect for the target bearing.

Initially, the DANN model is utilized for domain adaptation, reducing the difference between source domain data and target domain data. The innovative point of this method is aligning the extracted features with pre-trained features containing complete degradation information, making these extracted features domain-discriminative and invariant, while also encompassing comprehensive degradation information.

Finally, due to the structural similarity between the PRONOTIA bearing test platform of the FEMTO-ST Institute and the test structure of the ship propulsion shaft system, both of which are composed of equipment such as a motor, gearbox, bearings, coupling, and propeller (load), this common dataset is used to validate the effectiveness of the proposed algorithm. The experimental results show that compared with the MK-MMD algorithm, the proposed method achieved an average error reduction of 0.0626, and compared with the SA ConvLSTM algorithm, the average error was reduced by 0.0845. Additionally, the correlation coefficient R² indicates that the prediction results of this method are more reliable. Therefore, this method can be attempted to solve the problem of predicting the bearing life under multiple operating conditions in ship propulsion systems, as well as other similar cross-condition prediction scenarios.