Adversarial domain adaptation with classifier alignment for cross-domain intelligent fault diagnosis of multiple source domains

A single-source domain adaptation problem under diverse operating conditions has been widely studied [37]. In this study, the multi-source unsupervised domain adaptation method is proposed for cross-domain fault diagnosis. The illustration of single-source and multi-source domain adaptations is shown in figure 1. It can be seen that the diagnosis accuracy of the target domain can be improved through the organic combination of multiple source domains.

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Let D^sj = $\{{x}_i^{{sj}}$, ${y}_i^{{sj}}\}^{i=1}_{nsj}$ i = 1, 2, ..., n_sj, j= 1, 2, ..., N_s denote the labeled source samples, where ${x}_i^{{sj}}$ indicates the sample features of N_input dimensions from source domain j, y_sj represents the corresponding labels from source domain j, n_sj indicates the number of labeled samples of domain j and N_s indicates the number of the source domains. Let D^t = {D^t _train, D^t _test} denote the target domain samples; D^t _train and D^t _test indicate target domain training samples and testing samples, respectively. Among them, D^t _train = $\{{x}_i^{\text{train}}\}_{i=1}^{n_{\text{train}}}$, i = 1, 2,...,n_train denote the unlabeled target samples, where x^train _i indicates the sample features, and n_train indicates the number of unlabeled samples. Correspondingly, D^t _test = $\{{{x}_i^{\text{test}}, {y}_i^{\text{train}}}\}_{i=1}^{n_{\text{test}}}$, i = 1, 2,...,n_test denote the labeled target samples, where x_i ^test indicates the sample features, y_i ^test represents the corresponding labels, and n_test indicates the number of labeled samples. The purpose of the study is to train a model that can accurately diagnose D^t _test through D^sj and D^t _train.

The architecture overview and basic network of the proposed method are shown in figure 2, which includes feature extractor G, domain discriminator D and classifier C. The G receives multiple source domain samples and target domain samples simultaneously, and the one-dimensional feature vectors x= F(x) are obtained. After the feature vectors, all domain samples are fed into the D, which aims to align feature distributions of all domains. Specifically, the D learns domain-invariant features using adversarial learning between D and G by adding a simple gradient reversal layer (GRL). After the feature vectors, the classifier $\{C_j\}^{N_s}_{j=1}$ is added. All classifiers have a uniform network structure and initialization parameters. Each classifier C_j receives F(x) coming from jth source domain samples and target domain samples simultaneously and outputs the corresponding prediction label.

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• (a)  
  Source domain classification loss term

The multi-source domain fault diagnosis task is studied in this paper. Each set of source domain data x^sj corresponds to a classifier C_j . The parameters of the G and the C_j are updated using corresponding source domain samples. The cross-entropy loss is adopted in this term [38]. The cross-entropy loss of the all classifier C_j on labeled source samples $\{{{x}_i^{{sj}}}{{y}_i^{sj}}\}_{i=1}^{n_{sj}}$ is formulated as:

$$\begin{equation}\mathcal{L}_{_{cls}}^{} = \frac{1}{{{N_s}}}\sum\limits_{j = 1}^{{N_s}} {\mathcal{L}_{_{cls}}^j({C_j}(F({x^{sj}})),{y^{sj}})} \end{equation} \tag{ 1 }$$

where N_s indicates the number of all the source domains, ${\mathcal{L}}^j_{\text{cls}}$(·) indicates the cross-entropy loss of jth classifier, and ${\mathcal{L}}^j_{\text{cls}}$(·) is defined as follows:

$$\begin{align}\mathcal{L}_{_{cls}}^j({C_j}(F({x^{sj}})),{y^{sj}}) & = - {\mathbb{E}_{({x^{sj}},{y^{sj}}) \in {D^{sj}}}}\bigg[\sum\limits_{k = 1}^k {\mathbb{I}_{\left[k = {y^{sj}}\right]}} \nonumber\\ & \quad \log ({C_j}(F({x^{sj}}))) \bigg].\end{align} \tag{ 2 }$$

Here C_j (F(x^sj )) denotes the prediction probability output of x^sj by classifier C_j, and k indicates the number of health conditions.

The parameters of feature extractor, domain discriminator and classifier are defined as θ_G, θ_D and θ_Cj , while the parameters of all classifiers are defined as θ_C . For each classifier, the cross-entropy loss is minimized to seek optimal parameters of θ_G and θ_C , which can be expressed as follows:

$$\begin{equation}({\hat \theta _G},{\hat \theta _C}) = \mathop {\arg \min }\limits_{{\theta _G},{\theta _{{C_{^1}}}},{\theta _{{C_{^2}}}},\ldots,{\theta _{{C_{^{{N_c}}}}}}} \mathcal{L}_{_{cls}}^{}.\end{equation} \tag{ 3 }$$

• (b)  
  Domain distribution alignment loss term

To achieve the first domain distribution alignment stage, the domain discriminator D is introduced into the training process. There are N_s + 1 neurons in the final output layer of D, and each neuron represents a specific domain. The cross-entropy loss is also used in this term. Therefore, the domain classification loss can be defined as:

$$\begin{equation}\mathcal{L}_{_d}^{} = - {\mathbb{E}_{(x,y) \in {D^{sj}},D_{^{train}}^t}}\left[\sum\limits_{k = 1}^{1 + {N_s}} {{\mathbb{I}_{[y = k]}}\log (D(F(x)))} \right]\end{equation} \tag{ 4 }$$

where D(F(x)) denotes the prediction probability output of x by D.

In the training process, the distribution shift in the shared feature space F(x) is reduced through an adversarial learning process. First, the purpose of D is to accurately distinguish all domain samples. On the contrary, the purpose of G is to confuse the D. To be specific, the parameters of the D are updated to minimize ${\mathcal{L}}$ _d and the parameters of the G are updated to maximize ${\mathcal{L}}$ _d simultaneously using all the domain samples, which attempts to align domain distribution discrepancy [39]. Thus, the optimal parameters can be obtained by training this function:

$$\begin{equation}\begin{gathered} ({{\hat \theta }_D}) = \mathop {\arg \min }\limits_{{\theta _D}} \mathcal{L}_d^{} \hfill \\ ({{\hat \theta }_G}) = \mathop {\arg \max }\limits_{{\theta _G}} \mathcal{L}_d^{} \hfill \\ \end{gathered} .\end{equation} \tag{ 5 }$$

In this scenario, the parameters of equation (5) cannot be directly optimized because ${\mathcal{L}}$ _d is maximized by the G and ${\mathcal{L}}$ _d is minimized by the D simultaneously. To solve the problem, the GRL is introduced as shown in figure 2. GRL acts on the back propagation process of the network, and the sign of gradient is flipped after the gradient passes through the GRL [40]. This procedure ingeniously solves the issue that the maximum and minimum gradient cannot be trained simultaneously. In ADACL, the GRL is introduced between the G and the D. To be specific, the GRL can be written as a function R(x):

$$\begin{equation}\begin{gathered} R(x) = x \hfill \\ \frac{{{\text{d}}R(x)}}{{{\text{d}}x}} = - \lambda {\text{I}} \hfill \\ \end{gathered} \end{equation} \tag{ 6 }$$

where λ indicates the penalty coefficient and λ = 1 in the proposed method, and I denotes an identity matrix. In this way, equation (5) can be formulated as:

$$\begin{equation}({\hat \theta _G},{\hat \theta _D}) = \mathop {\arg \min }\limits_{{\theta _G},{\theta _D}} \mathcal{L}_d^{}.\end{equation} \tag{ 7 }$$

• (c)  
  Domain classifier alignment loss term

The domain distribution alignment can only achieve the confusion of the multi-domain features. Therefore, the target domain samples near the decision boundary are easily misclassified by the classifier. Intuitively, for these target samples close to the decision boundary, different classifiers are likely to make different predictions. In this case, the number of misclassified samples close to the decision boundary is reduced by reducing the predicted disagreement of classifiers.

To be specific, all classifiers are used to construct a discriminator, and the L1-norm of all the classifier predictions of target train samples x^train as discrepancy loss:

$$\begin{equation}\mathcal{L}_{_{dis}}^{} = {\mathbb{E}_{x \in D_{^{train}}^t}}\frac{1}{{{N_s}}}\sum\limits_{j = 1}^{{N_s} - 1} {\sum\limits_{i = j + 1}^{{N_s}} {{{\left\| {{C_j}(F({x^{train}})) - {C_i}(F({x^{train}}))} \right\|}_1}} } \end{equation} \tag{ 8 }$$

where C_j (F(x^train)) and C_i (F(x^train)) denote the prediction probability output of x^train by classifier C_j and C_i . In the same way, the optimal parameters can be obtained by training this function:

$$\begin{equation}({\hat \theta _G},{\hat \theta _C}) = = \mathop {\arg \min }\limits_{{\theta _G},{\theta _{{C_{^1}}}},{\theta _{{C_{^2}}}},\ldots,{\theta _{{C_{^{{N_c}}}}}}} \mathcal{L}_{_{dis}}^{}.\end{equation} \tag{ 9 }$$

• (d)  
  Overall formulation of the proposed method

To sum up, the proposed method includes three loss terms. All the terms are integrated to obtain the following overall optimization objective:

$$\begin{equation}\mathcal{L} = \mathcal{L}_{_{cls}}^{} + \alpha \mathcal{L}_{_d}^{} + \beta \mathcal{L}_{_{dis}}^{}\end{equation} \tag{ 10 }$$

where ${\mathcal{L}}$ denotes total loss, and α and β are two trade-off parameters.

Specifically, the G is optimized to minimize the ${\mathcal{L}}$ _cls, ${\mathcal{L}}$ _d and ${\mathcal{L}}$ _dis. The D is optimized to minimize the ${\mathcal{L}}$ _d . The C is optimized to minimize the ${\mathcal{L}}$ _cls and ${\mathcal{L}}$ _dis. Thus, the overall optimization problem is as follows:

$$\begin{equation}\begin{gathered} ({{\hat \theta }_G}) = \arg \{ \mathop {\min }\limits_{{\theta _G}} \mathcal{L}_{_{cls}}^{},\mathop {\min }\limits_{{\theta _G}} \mathcal{L}_d^{},\mathop {\min }\limits_{{\theta _G}} \mathcal{L}_{_{dis}}^{}\} \hfill \\[3pt] ({{\hat \theta }_D}) = \mathop {\arg \min }\limits_{{\theta _D}} \mathcal{L}_d^{} \hfill \\[3pt] ({{\hat \theta }_C}) = \arg \{ \mathop {\arg \min }\limits_{{\theta _{{C_{^1}}}},{\theta _{{C_{^2}}}},\ldots,{\theta _{{C_{^{{N_c}}}}}}} \mathcal{L}_{_{cls}}^{},\mathop {\arg \min }\limits_{{\theta _{{C_{^1}}}},{\theta _{{C_{^2}}}},\ldots,{\theta _{{C_{^{{N_c}}}}}}} \mathcal{L}_{_{dis}}^{}\} \hfill \\ \end{gathered}. \end{equation} \tag{ 11 }$$

The algorithm of the ADACL is summarized in Algorithm 1. During the training process, we feed uniform numbers of source samples and target samples to the network to calculate the three losses simultaneously, and minimize total loss by stochastic gradient descent (SGD) until the maximal epoch is met [41]. Finally, to test the network performance on the target domain testing samples, the average of all classifier outputs is taken as the final output.

Algorithm 1 Training procedure of ADACL

Input: Labeled source domain samples Dsj = $\{{{x}_i^{{sj}}}, {{y}_i^{sj}}\}_{i=1}^{n_{sj}}$, Unlabeled target domain samples Dt train = $\{{x}_i^{\text{train}}\}_{i=1}^{n_{\text{train}}}$, and the number of training epoch E.
Output: Configurations of ADACL
1: Initialize θG, θD and θC
2: for e = 1 to E do
3:	Uniformly sample (xs, ys ) from Dsj
4:	Uniformly sample (xtrain) from Dt train
5:	Gradually change α and β from 0 to 1
6:	Feed the Dsj and Dt train to the G, D and C
7:	Calculate ${\mathcal{L}}_{\text{cls}}$ using equation (1)
8:	Calculate ${\mathcal{L}}_{{d}}$ using equation (4)
9:	Calculate ${\mathcal{L}}_{\text{dis}}$ using equation (8)
10:	Optimize multiple subnetworks G, C, and D in turn:
11:	$({\hat \theta _G}) = \arg \{ \mathop {\min }\limits_{{\theta _G}} \mathcal{L}_{_{cls}}^{},\mathop {\min }\limits_{{\theta _G}} \mathcal{L}_d^{},\mathop {\min }\limits_{{\theta _G}} \mathcal{L}_{_{dis}}^{}\}$
12:	$({\hat \theta _D}) = \mathop {\arg \min }\limits_{{\theta _D}} \mathcal{L}_d^{}$
13:	$({\hat \theta _C}) = \arg \{ \mathop {\arg \min }\limits_{{\theta _{{C_{^1}}}},{\theta _{{C_{^2}}}},\ldots,{\theta _{{C_{^{{N_c}}}}}}} \mathcal{L}_{_{cls}}^{},\mathop {\arg \min }\limits_{{\theta _{{C_{^1}}}},{\theta _{{C_{^2}}}},\ldots,{\theta _{{C_{^{{N_c}}}}}}} \mathcal{L}_{_{dis}}^{}\}$
14: end for

In the network architecture, some effective operations [42, 43], namely convolution (Conv), batch normalization, rectified linear unit and maximum pooling (MP) are introduced to improve the ability of feature extraction. The architecture and parameters of the ADACL are displayed in table 1, where k indicates the number of health conditions and N_s indicates the number of the source domains. As the vibration signal is one-dimensional data, one-dimensional CNN is adopted. The raw signal is inputted into the G directly without manually extracting features, which avoids heavy manual signal processing and improves the application ability of the industry. The length of each sample is 2048.

In network training, the optimization objective is optimized via the SGD with a momentum of 0.9. The epoch and batch size of training processes are set at 500 and 50, respectively. The initial learning rate is 0.1, and its value decreases by 10% after every 100 epochs. The proposed network is implemented in the PyTorch framework. In calculating the total loss, α and β are gradually changed from 0 to 1 to suppress noisy signals in the initial stage of the training process, and they can be calculated as [40]:

$$\begin{equation}{\alpha _i} = {\beta _i} = \frac{2}{{1 + \exp ( - \gamma (\frac{i}{{\text{E}}}))}} - 1\end{equation} \tag{ 12 }$$

where i indicates ith epoch in the training, E indicates the number of the epoch, and γ is 10 throughout the experiments.

To assess the effectiveness of the ADACL comprehensively, several common methods are implemented as comparisons. The network structure and parameters of these comparison approaches are similar to the proposed method, and they are implemented on the same dataset.

• (a)  
  Without domain adaptation (WDA).

First, a simple method WDA is carried out, where the model is trained only using labeled source domain samples. In this approach, the network structure includes a feature extractor and a classifier, and cross-entropy loss is adopted as an optimization objective. Since the multi-source domain scenario is studied in this paper, WDA can be divided into two schemes: 1. WDA_high: with one source domain as the training data, WDA_high represents the best single source domain diagnosis results in multiple source domains. 2. WDA_com: WDA_com represents the diagnosis results with all source domains are combined as the training data.

• (b)  
  MMD-based domain adaptation method.

In this approach, the network structure includes a feature extractor, an MMD adaptation layer and a classifier, and the MMD adaptation layer is added behind the feature extractor. The MMD is used to minimize the distribution discrepancy of the two domains. Hence, a cross-entropy loss and MMD loss are adopted as the optimization objective in this comparison approach. In the same way, because of the existence of multiple source domains, MMD_high is defined as the best single source domain diagnosis in multiple source domains.

• (c)  
  Domain adversarial network (DAN)-based domain adaptation method

Considering that the proposed method is an extension of the adversarial method, the original domain adversarial method is used as a comparison approach. In this approach, the network structure is consistent with the ADACL. The cross-entropy loss and domain adversarial loss are adopted as optimization objectives. Similarly, DAN_high is defined as the best single source domain diagnosis in multiple source domains. DAN_com is defined as the diagnosis results in all source domains combined as the training data, which also represents the proposed method without considering the domain classifier alignment loss.

3.3.1. Experimental setup.

In this study, a practical bearing test-rig is constructed first to assess the effectiveness of the ADACL. In this experiment, the accelerometers used to collect vibration signals are installed on the bearing house, and the vertical vibration signals are used for fault analysis. This experiment considers five health conditions i.e. normal (N), outer-race fault (OF), inner-race fault (IF), ball fault (BF) and compound fault (OF_BF). The bearing test-rig and three types of fault are shown in figure 3. The experiment is implemented under three rotating speed conditions, namely 600 rpm (R₁), 1200 rpm (R₂) and 1800 rpm (R₃). The sampling time is set at 220 s, and the sampling frequency is 20 kHz.

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The waveforms of all health conditions under the three operating conditions are shown in figure 4. From the vibration waveforms, in the normal condition, no obvious impacts appear in various operating conditions. However, in the four fault conditions, there are some distinctions between the transient fault characteristics of different operating conditions. The generalization ability of the deep learning model is seriously weakened due to domain shift caused by the feature distribution discrepancy. In existing studies, the domain adaptation technique can be utilized to shorten the domain distribution discrepancy.

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In this experiment, three operating conditions (R₁, R₂ and R₃) are studied. According to the multi-source domain adaptation scenario in this study, three domain adaptation tasks are designed. That said, the model is trained with two source domains and a target domain. The details of the three domain adaptation tasks are shown in table 2.

Table 2. The details of three tasks on the bearing test-rig dataset.

	The operating condition		Number of samples
	Source domain	Target domain	Source	Target	Target
Task	(rpm)	(rpm)	training	training	testing	Fault categories
R1, R2→R3	600, 1200	1800	2 × 5 × 400	5 × 400	5 × 200
R1, R3→R2	600, 1800	1200	2 × 5 × 400	5 × 400	5 × 200	Five conditions (labels 0–4)
R2, R3→R1	1200, 1800	600	2 × 5 × 400	5 × 400	5 × 200

3.3.2. Experimental results.

The diagnostic performance on the bearing test-rig dataset is analyzed. The fault diagnostic results of all methods in three domain adaptation tasks are given in table 3. The average diagnostic accuracy rate of the proposed method is 90.7%, which is the best diagnostic accuracy rate among all diagnostic methods. This sufficiently demonstrates that the ADACL can effectively achieve cross-domain fault diagnosis under diverse operating conditions, and its results are better than existing methods.

Table 3. Fault classification accuracy on the bearing test-rig dataset.

	Fault classification accuracy (%)
Method	R1, R2→R3	R1, R3→R2	R2, R3→R1	Average
WDAhigh	65.6	65.8	32.5	57.6
WDAcom	81.3	79.2	56.2	72.2
MMDhigh	90.2	89.1	62.8	80.7
DANhigh	89.1	89.3	69.3	82.9
DANcom	90.4	91.9	71.5	84.6
Proposed	94.8	99.6	77.7	90.7

To be specific, some conclusions can be drawn. The WDA_com of combining two source domains as a training source domain is generally better than using a single source domain method on the diagnosis result. This may be due to the fact that the addition of discrepant samples expands the generalization ability of the network, thereby improving the accuracy of diagnosis. As can be seen from the results of MMD_high and DAN_high, in most domain adaptation tasks, encouraging results can be obtained by learning the domain-invariant representation of source domain and target domain. In DAN_com, compared with single source adversarial learning, the diagnostic performance is not significantly improved only by introducing multiple source domains into the DAN. However, adding domain classifier alignment on the basis of multi-source adversarial learning can narrow the gap among all classifiers and improve the classification performance of all classifiers. Hence, these results indicate that the ADACL can effectively achieve cross-domain fault diagnosis in two source domains, although the operating conditions vary greatly among all domains.

To further show the advantage of the ADACL, the t-distributed stochastic neighbor embedding technique (t-SNE) [44] is used to implement two-dimensional feature visualization of output features. Taking the task R₁, R₃→R₂ as an example, the two-dimensional visualization results of the training process are shown in figure 5. It is clear that the features of two source domain samples and target domain training samples are well separated, and the feature distributions of the same health conditions are aligned. Figure 6 shows the visualization results of the testing process for all methods. As shown in figure 6, the feature map by WDA_high does not cluster well in health conditions N, IF and OF, because the three health conditions may present similar features under different operating conditions. The clustering conditions are improved in health conditions N, IF and OF by introducing the general domain adaptation method, but there is still a slight overlap between IF and OF. However, it is clear that the learned feature by ADACL achieves the best clustering and separability in all health conditions.

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To present detailed diagnostic results for each health condition by different methods in task C₁, C₃→C₂, the confusion matrices of the six methods on the bearing test-rig dataset are given in figure 7. In the confusion matrices, the predicted label and the ground-truth label are displayed, and the prediction accuracy rates and prediction error rates for each health condition are also displayed. It is clear from figure 7 that the ADACL can achieve accurate predictions of all health conditions on the bearing test-rig dataset.

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Considering that there may be a variety of noises in the actual industrial scenario, the diagnostic performance is evaluated under the noisy environment in the task C₁, C₃→C₂ to verify the generalization ability of the proposed method. In this experiment, Gaussian noise with signal-to-noise ratios (SNR) of −4, 0, 4, 8 are added to the source domain signal and the target domain signal, where the SNR is defined as [45]:

$$\begin{equation}{\text{SNR}}({\text{db}}) = 10{\log _{10}}({P_{signal}}/{P_{noise}})\end{equation} \tag{ 13 }$$

where P_signal and P_noise indicate the powers of the original signal and the added noise, respectively.

Figure 8 shows the diagnostic results of the proposed method and comparison methods under different SNR on the bearing test-rig dataset. It can be seen that the environmental noise significantly reduces the diagnostic performance of the model in methods WDA_high, MMD_high and DAN_com. However, in the proposed method and the WDA_com method, environmental noise has little effect on diagnostic performance. Since the introduction of multi-source domains improves the generalization ability of the model, the performance of the model against noise is enhanced. Therefore, the proposed method can also obtain accurate prediction results in a noisy environment.

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3.4.1. Experimental setup.

The generic gearbox dataset is analyzed as the second case. The generic gearbox dataset is from the 2009 challenge data of Prognostics and Health Management [46]. The inside details of the gearbox are shown in figure 9. The main elements of the generic gearbox consist of four gears, three axes (input axis IS, intermediate axis ID, output axis OS) and six bearings (three on input side: IS and three on output side: OS). This dataset considers eight health conditions, as shown in table 4. Four rotating speeds are considered in this study, namely 35 Hz (R₁), 40 Hz (R₂), 45 Hz (R₃) and 50 Hz (R₄). The vibration signals are obtained from the output shaft end by an accelerometer with a sampling frequency of 66.67 kHz.

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Table 4. Pattern label description of the generic gearbox.

	Gear				Bearing						Shaft
Label	32 T	96 T	48 T	80 T	IS:IS	ID:IS	OS:IS	IS:OS	ID:OS	OS:OS	Input	Output
0	G	G	G	G	G	G	G	G	G	G	G	G
1	C	G	E	G	G	G	G	G	G	G	G	G
2	G	G	E	G	G	G	G	G	G	G	G	G
3	G	G	E	Br	B	G	G	G	G	G	G	G
4	C	G	E	Br	In	B	O	G	G	G	G	G
5	G	G	G	Br	In	B	O	G	G	G	Im	G
6	G	G	G	G	In	G	G	G	G	G	G	Ks
7	G	G	G	G	G	B	O	G	G	G	Im	G

G: good; C: chipped; E: eccentric; Br: broken; B: ball; In: inner race; O: outer race; Im: imbalance; Ks: keyway sheared.

In this experiment, four operating conditions (R1, R2, R3 and R4) are studied. In the first case, the proposed method is validated in a scenario with two source domains. Therefore, in this experiment, a scenario with three source domains was studied. Four domains can implement four domain adaptation tasks; the details of the four domain adaptation tasks are shown in table 5.

Table 5. The details of the four tasks on the generic gearbox dataset.

	The operating condition		Number of samples			Fault categories
Task	Source domain (Hz)	Target domain (Hz)	Source training	Target training	Target testing
R1, R2, R3→R4	35, 40, 45	50	3 × 8 × 100	8 × 100	8 × 100	Eight conditions
R1, R2, R4→R3	35, 40, 50	45	3 × 8 × 100	8 × 100	8 × 100	(labels 0–7)
R1, R3, R4→R2	30, 45, 50	40	3 × 8 × 100	8 × 100	8 × 100
R2, R3, R4→R1	40, 45, 50	35	3 × 8 × 100	8 × 100	8 × 100

3.4.2. Experimental results.

The diagnostic performance on the generic gearbox dataset is analyzed in three source domain scenarios. The statistics of the diagnostic results of all methods in four domain adaptation tasks are given in table 6. Similar to the case study in the bearing test-rig dataset, the results of the ADACL are significantly better than those of the comparison methods. This sufficiently demonstrates that the proposed method can effectively achieve cross-domain fault diagnosis under three source domains.

Table 6. Fault classification accuracy on the generic gearbox dataset.

	Fault classification accuracy (%)
Method	R1, R2, R3→R4	R1, R2, R4→R3	R1, R3, R4→R2	R2, R3, R4→R1	Average
WDAhigh	76.3	78.6	80.2	79.3	78.6
WDAcom	85.7	87.8	88.4	87.3	87.3
MMDhigh	88.6	90.3	91.5	90.1	90.1
DANhigh	91.6	91.1	90.7	90.5	91.0
DANcom	93.8	95.6	94.9	92.3	94.2
Proposed method	96.3	97.9	97.3	96.8	97.1

Similarly, to visually demonstrate the superiority of the ADACL, taking the task R₁, R₂, R₄→R₃ as an example, the visualization results of the training process are shown in figure 10. It shows that the four domain data are well separated, and the features of the four domains are well aligned. The visualization results and confusion matrices of the testing process for all methods on the task R1, R2, R4→R3 are also given in figures 11 and 12. It is clear that some areas are overlapping between diverse health conditions by the compared methods. Meanwhile, the features of all categories by the proposed method are well separated, which means that the ADACL can correctly distinguish between diverse health conditions.

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Figure 13 shows the diagnostic results of the proposed method and comparison methods under different SNR on the task R1, R2, R4→R3. It is clear that the diagnostic performance of the proposed method is less affected by environmental noise. Therefore, the proposed method also has a strong ability to resist noise in the case of three source domains. To sum up, case 2 further illustrates that the proposed method can also accurately achieve cross-domain fault diagnosis in the case of three source domains.

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