Cyclostationarity-enhanced multi-source domain generalization with mixture-of-experts for rolling bearing fault diagnosis

2.1. Multi-source domain generalization for fault diagnosis

Multi-source domain generalization is a rising perspective for addressing the generalization challenge in complex industrial environments of rotational mechanical systems by utilizing data from multiple source domains. Methods based on data augmentation aim to create data of pseudo domains assisted with source data. Typical methods include direct numerical operations, e.g., interpolation [36], and distribution-based combination, e.g., convex combination with Dirichlet-based Mixup [7]. Moreover, Wang et al. [12] proposed a collaborative domain-cycling framework with multiple feature domains and employed multiple metrics to filter high-quality data. However, the semantics evaluation of the augmented data is a challenging threshold for further investigation. Zheng et al. [37] proposed to use Local Fisher Discriminant Analysis (LFDA) to first obtain the optimal discriminant structure of each source domain, and then map the general mean subspace onto the Grassmann manifold as a general feature mapping transformation kernel for data from different domains. Explicit feature distribution alignment is a popular regularization view to constrain the model to learn general features from multiple source domains, e.g., MMD loss [38], triplet loss [39], and central loss [40]. To exploit the advantages of each source domains, Zhao et al. [16] trained domain-specific branches for each source domain with a similarity indicator branch to have a comprehensive evaluation. Gao et al. [19] proposed a meta-learning strategy to simulate the generalization scenario of industrial processes with meta-test sub-domains divided from source data to enhance learning general representation. Mu et al. [17] proposed a Theil index-based meta-learning network (TTIMN-GCS), where a task-orientated Theil index is designed to balance the inequality among meta-tasks to achieve a more generalized meta-optimization.

2.2. Causality-inspired domain generalization for fault diagnosis

Causality-inspired domain generalization methods can be divided into two groups: feature separation and feature purification. Feature separation methods typically construct a structural causal model (SCM) based on the data-generating process, followed by regularization based on do-calculus to extract causal variables [30, 26, 31]. Guo et al. [23] proposed a causal independence and sparse shift network (CIS2N), which samples intervention pairs with the same label from different domains and is trained with a causal independence loss based on independent causal mechanisms (ICM) and a sparse shifts loss based on sparse mechanism shifts (SMS). Jia et al. [29] found that solely taking causal factors into account leads to insufficient removal of the spurious correlation. To address this, they proposed a deep causal factorization network (DCFN), which adds a domain classification flow to extract non-causal factors. They [25] later extended it to a causal disentanglement domain generalization model (CDDG) by introducing generative modeling with a reconstruction loss. Another line of work is based on invariant risk minimization (IRM) and its variants, which treats data from each environment individually and learns an optimal classifier on top of a data representation matching all environments [41]. Mo et al. [42] proposed a sparsity-constrained invariant risk minimization method (SCIRM), which introduces differentiable sparsity regularizations to learn invariant classifiers across domains with sparse and effective features. The above works can be classified as class-conditioned methods, focusing on the invariant distribution of causal features conditioned on health state. However, Cheng et al. [22] noted that the distribution of causal features with the same label may shift due to the variation of bearings. Inspired by causal matching [43], they proposed a three-stage based domain generalization fault diagnosis method (LOODG) with an object-conditional domain invariant rule to align features of samples from different working conditions but similar bearing. In addition to regularization constraints-based approaches, Li et. al. [21] proposed Whitening-Net, which employs layer normalization (LN) to stabilize the distribution of individual sample and instance normalization (IN) to eliminate domain specific features. For models established upon feature purification, Wu et al. [4] proposed an adversarial-causal representation learning network (ACRLN) with spatial mask domain adversarial strategy using a gradient reversal layer and an auxiliary branch to remove domain specific features. Ma et al. [27] employed gradient truncation with the guidance of a binary mask created by domain discriminators to explicitly discard non-causal features at the layer and channel level. Besides, much research has employed masking mechanism guided by specific loss functions to purify causal features [44, 45]. Despite these advances, most existing domain generalization methods operate directly on the entire signal from an implicit perspective and rely heavily on neural network autonomy, which result in limited transparency and retention of spurious features.

3.1. Problem definition

The domain generalization problem for rolling bearing fault diagnosis is set to recognize the health state of rolling bearing under unseen working conditions by acquiring generalizable knowledge from labeled data under multiple working conditions. In this context, let $\{D^s{\}}_{s=1}^S$ denotes $S$ source domains with each domain $D^s=\{x_n^s,y_n^s{\}}_{n=1}^N$ containing $N$ labeled data, where $x_n^s\in {\mathbb{R}}^{1\times L}$ represents a vibration sample of length $L$ and $y_n^s\in {\mathbb{R}}^{1\times C}$ is its corresponding label with $C$ health categories. Let $\{D^t{\}}_{t=1}^T$ denotes $T$ unseen domains with each domain $D^t=\{x_m^t,y_m^t{\}}_{t=1}^M$ containing $M$ unlabeled data of the $t$-th unseen target domain. Due to the distribution shift under different working conditions, e.g. speed and load, the data distributions differ among source domains and between source and target domains, i.e., $P_s\ne P_{s\text{'}}$ with $s\ne s\text{'}$, and $P_t\ne P_s$, where $s$, $s\text{'}\in \{1,\cdots ,S\}$. The objective is to make full use of the source data to develop and optimize a method with good generalization to unseen domains.

3.2. Cyclostationarity-enhanced representation of frequency band

An effective representation is the core of a data-driven model. To establish a transparent diagnostic model, we choose to feed a representation based on cyclostationarity with spectral correlation density (SCD) which is a high-dimensional characteristic with the ability to reveal fault modulation frequencies.

– Frequency band decomposition with wavelet packets. First, to realize from the perspective of frequency bands, an input univariate time series $x$ is decomposed into $P$ patches $x_{1:P}=\{x_1,\cdots ,x_P\}\in {\mathbb{R}}^{P\times L_W}$ corresponding to wavelet coefficients spanning from low-frequency to high-frequency bands, where $L_w$ is the length of wavelet coefficients. With a $l$ level wavelet packet decomposition, the input univariate time series $x$ is decomposed into $2^l$ patches, and patch $x_i^l\in R^{1\times L_w}$ captures the features of analyzed signal in the frequency range of $\left[i\times f_s/2^{l+1},\left(i+1\right)\times f_s/2^{l+1}\right]$. Each pair of wavelet coefficients at $l$ level $x_{2i}^l$ and $x_{2i+1}^l$ are calculated from $x_i^{l-1}$ through low-pass filter $h\left[k\right]$ and high-pass filter $g\left[k\right]$ respectively as:

The two filters are constructed with a scaling function $\varphi \left(t\right)$ and a selected primary wavelet function $\psi \left(t\right)$ as:

where ${\varphi }_{-1,k}\left(t\right)=\sqrt{2}\varphi \left(2t-k\right)$, which is the scaling function at a finer scale, $〈\cdot 〉$ is the inner product operator. Note that the two filters are mutually orthogonal with relationship of $g\left[k\right]={\left(-1\right)}^{k}h\left[1-k\right]$ so that filtered signals through them are independent and represent low frequency band and high frequency band respectively.

– Cyclostationarity-enhanced representation. For the sake of effectiveness and interpretability, we choose spectral correlation density as the input feature, as it can effectively characterize the hidden periodic modulation mechanisms caused by different localized faults of rolling bearing through spectral correlation peaks, which provides a direct visualization of fault-induced modulation. As a classical feature of second order cyclostationarity, spectral correlation density is calculated based on the autocorrelation $R_x\left(t,t-\tau \right)=E\left\{x\left(t\right)x\left(t-\tau \right)\right\}$, where $\tau$ is the time shift. As the periodicity, it can be expanded into a Fourier series with cycle frequencies $\alpha$ as:

where $R_x^{\alpha }\left(\tau \right)=\frac{1}{T}{\int }_0^T{R}_x\left(t,t-\tau \right)e^{-j2\pi \alpha \left(t-\frac{\tau }{2}\right)}dt$ is defined as the cyclic autocorrelation function of cycle frequency $\alpha$, indicating modulation strength of cycle frequency $\alpha$. Then, it can be calculated by:

Then, the spectral correlation density can be calculated by the Fourier transform of the cyclic autocorrelation function with respect to the time shift $\tau$ as:

Based on Eq. (5), with the patches $x_{1:P}$, we obtain the cyclostationarity-enhanced representation $h_{1:P}\in {\mathbb{R}}^{P\times C\times S}$, where $C$ denotes the cycle frequency dimension, $S$ denotes the spectral frequency dimension. Each element of $h_i$ indicates the modulation strength of cycle frequency $\alpha$ on spectral frequency $f$.

3.3. Cycle frequency-level adaptive feature extraction

– Cycle frequency-level feature extraction via MoE with multi-view router. The adaptive feature extractor is designed based on a mixture-of-experts (MoE) block, which is a block of dynamically routed expert networks with each expert implemented by a feed-forward network (FFN). Specifically, input with the cyclostationarity-enhanced representation $h_{1:P}$, the feed forward process is expressed as:

where $\mathrm{L}\mathrm{N}\left(\cdot \right)$ denotes layer normalization used to stabilize the instance distribution, $f_{MoE}\left(\cdot \right)$ denotes the mixture-of-experts module which is given by:

where $R\left(\cdot \right)$ is the router of the gate $G\left(\cdot \right)$ calculating logits for assigning features $h$ to different experts, $\mathrm{T}\mathrm{O}{\text{P}}_{\text{k}}\left[\cdot \right]$ is an operation to only activate the top $k$ experts for each feature, specifically, it sets all other elements in the output vector as zero except for the elements with the largest $k$ logits, $\mathrm{F}\mathrm{F}\mathrm{N}\left(\cdot \right)$ denotes an expert combined with a fully-connected neural network and a nonlinear activation function, $E$ and $k$ are hyperparameters representing the number of experts and the number of selected top experts respectively.

For the routing scheme, [35] shows that the cosine router achieves better performance than the linear router in domain generalization tasks because of its ability to mitigate the representation collapse issue. Specifically, the cosine router is calculated through cosine similarity among features and experts. Typically, the routing score for each token is computed as:

where $S_E\in {\mathbb{R}}^{D_{E,D}\times E}$ represents the one-dimensional feature space for each expert, which assumes that each expert is responsible for a group of attributes with similar semantic. However, unlike visual tokens where each can represent specific attributes individually, e.g. ear, mouth, or leg, in this context where the tokens are cycle frequencies, the spectrum of single cycle frequency, i.e., the slice of the spectral correlation density function at a particular cycle frequency, can hardly exhibit clear semantic characteristics individually, e.g. fault, harmonic, or noise. The semantic of single cycle frequency commonly depends on both individual energy and the distribution structure of cycle frequencies [46, 47], which can be characterized by:

where $E\left({\alpha }_k\right)={\int }_{f_{min}}^{f_{max}}{\left|S^{{\alpha }_k}\left(f\right)\right|}^{2}df$ denotes the modulation energy of ${\alpha }_k$, $\mathcal{R}\left({\left\{{\alpha }_j\right\}}_j^J\right)$ denotes the structural context including harmonic relationship ${\alpha }_j={n\alpha }_k$, where $n$ represents the $n$-th harmonic order, and sideband relationship ${\alpha }_{j^{\text{'}}}={\alpha }_j\pm \delta f_{shaft}$, where the $f_{shaft}$ represents the rotational frequency. Moreover, to activate suitable and sufficient features for each cycle frequency of samples across different domains, we further evaluate the sample similarity to ensure that each expert focuses on cycle frequency with similar representation. To this end, we introduce a high-dimensional expert feature space as $S_E\in {\mathbb{R}}^{D_{E,P}\times D_{E,C}\times D_{E,D}\times E}$ to evaluate comprehensively in routing. Then, for the high-dimensional input feature $h$, the routing score is calculated from different dimensions, including spectral frequency (representing cycle frequency-level view), cycle frequency (representing frequency band-level view), and frequency band (representing sample-level view), which is expressed as:

where $S_E\in {\mathbb{R}}^{D_{E,P}\times D_{E,C}\times D_{E,D}\times E}$ is a learnable matrix representing experts’ features, $R_P\left(h\right)$, $R_C\left(h\right)$ and $R_S\left(h\right)$ refers to the routing with features of spectral frequency, cycle frequency and frequency band respectively, $\mathrm{M}\mathrm{E}\mathrm{A}{\text{N}}_{\left(\text{d}\right)}\left(S_E^T\right)$ denotes calculating the mean along dimension $d$ to get features of the rest dimension, $W_{\cdot }\cdot \mathrm{M}\mathrm{E}\mathrm{A}{\mathrm{N}}_{\left(\cdot \right)}\left(h\right)$ is the projection function for input features mapping to a hypersphere, ${\tau }_R$ is a learnable parameter used to adjust the sharpness of the probability distribution to control the routing assignments. Through the above feature extraction, we obtain the cycle frequency-level feature $z\in {\mathbb{R}}^{P\times C\times D}$.

– Auxiliary loss. To avoid the self-reinforcing imbalanced state that a gating network are prone to converge to, where a few experts are more frequently activated than others, followed [35], we optimize an importance loss ${\mathcal{L}}_{imp}$, calculated with the square of coefficient of variation (CV) of total routing scores of all features to each expert, to encourage balanced routing scores across experts, and a load loss ${\mathcal{L}}_{load}$, calculated with the square of coefficient of variation of total assignment probabilities of all features to each expert, to encourage balanced assignment to ensure the sufficient usage of all experts:

where $\mathrm{\Phi }$ is the cumulative distribution function of a Gaussian distribution, ${\eta }_k$ is the $k$-th largest routing score of each feature used as the threshold and only scores greater than it being assigned.

– Cycle frequency-level feature enhancement guided by prior knowledge. With the cycle frequency-level feature extracted by the MoE block, we employ a channel attention module (CAM) along the dimension of cycle frequency to highlight the fault-related cycle frequencies and enhance the corresponding features. The distribution of the fault-related cycle frequencies satisfies a sparsity property: the fault-related characteristic frequency and its harmonics are sparsely distributed at integer multiples of the fundamental frequency, while most other frequencies are irrelevant and redundant for diagnosis. However, the conventional Sigmoid layer produces smooth attention scores, which are insufficient for such sparsity. Inspired by [48], we redesign the channel attention with two branches: a squared ReLU-based branch that suppresses unrelated cycle frequencies with negative scores and propagates the most useful information flow forward, and a conventional Sigmoid-based branch that ensures sufficient information flow and keeps the model from falling into local optima during training. The process is expressed as:

where $\mathrm{M}\mathrm{A}\mathrm{X}\left(\cdot \right)$ denotes the maximum operation, $W_1$ and $W_2$ are the parameter of fully connected layers.

Although the above design can effectively highlight informative cycle frequencies, we observe performance degradation when transferring across a large span from high-speed to low-speed conditions. Through analyzing the attention scores, we find out that fault impacts under high-speed conditions can trigger strong high-order harmonics, which provide a simple solution for the model. However, these high-order harmonics become weaker under low-speed conditions as the dominance of random vibrations over fault impulses [49] and lead the model to mistakenly recognize fault states as normal. To address this issue, we introduce a regularization term on scores produced by the channel attention module to encourage the model to pay more attention on low cycle frequencies while still retaining the ability to learn the discriminative feature from high cycle frequencies. The regularization term is calculated by:

where $w_{z,i}$ denotes the average score of the $i$-th group cycle frequencies. This group-wise design is set to align with the discrete sparsity property of fault-related frequency distribution. The gradient of this regularization with respect to $w_{z,i}$ is derived as:

where $f_i^n={\sum }_{i=1}^{C-1}{e}^{{∆}_i^n}1\{{∆}_i^n\ge 0\}$. For each part, when ${∆}_i^n\ge 0$, the corresponding item becomes $-e^{{∆}_i^n}$ to $w_{z,i}^n$ and $e^{{∆}_i^n}$ to $w_{z,i+1}^n$, thereby enforcing a monotonic decrease between adjacent group scores during backpropagation. When ${∆}_i^n<0$, the corresponding item will be truncated to control the scale of $w_{z,i}^n$ and prevent gradient explosion.

Then, a global average pooling layer (GAP) is used to aggregate each cycle frequency as:

In this way, we obtain the enhanced cycle frequency-level feature $\tilde{z}\in {\mathbb{R}}^{P\times C}$.

3.4. Frequency band-level causal localization

From the perspective of considering frequency bands associated with environments as confounders, a frequency band-aware module is proposed to localize fault-related frequency bands and suppress the influence of environment-related frequency bands.

– Localization with self-attention. To implement classification with the extracted features of frequency bands, inspired by prior work on modeling multiple sub-information in the classification of time series [50], we adopt a tokenized Transformer based on the self-attention mechanism of frequency bands. The tokenized Transformer tends to be popular in computer vision and time series, since it has been proven in [50] that it can reduce the complexity of developing a good classifier by explicitly modeling the correlation among input elements and lowering the class-conditioned entropy, which measures the uncertainty of features given that the class label is known. This means that employing a class token to aggregate features of causal frequency bands through modeling their correlations lowers the difficulty of classification compared to directly feeding raw features of frequency bands into a classifier. Additionally, the attention mechanism helps generate relation intensity graphs to visualize the importance of each frequency band. Here, we first employ a learnable class token $z_{cls}\in R^{1\times C}$ to aggregate the features of causal frequency bands.

Specifically, we concatenate the class token with the features ${\tilde{z}}_{1:P}$ to yield a tokenized bag $Z_{cls}=\left[z_{cls},{\tilde{z}}_1,\cdots ,{\tilde{z}}_P\right]\in {\mathbb{R}}^{\left(1+P\right)\times C}$ and aggregate fault-related information to update $z_{cls}$ as:

where $W_F$ is the parameter of a fully connected layer used to fuse the output of multiple attention heads, $hea{d}_h$ denotes a self-attention head expressed as:

where $W_h^Q$, $W_h^K$ and $W_h^V$ are the parameters of the transformation layers. Nystrom attention [51] is used here to reduce the complexity of computation.

After the Transformer block, we only pass the class token to the classifier to make a prediction. To guide the learning of fault-related features and causal frequency band, the cross-entropy loss is used to perform classification with the prediction of classifier $G_c$ as:

where $y$ is the class label, $N$ is the batch size.

– Sparsity regularization for causal frequency bands. Beyond the tokenized Transformer, some spurious frequency bands may remain. Considering the difficulty in convergence and tuning of gradient-related methods, e.g., gradient reversal, and the obstacle of obtaining accurate domain labels, we purify causal frequency bands from another perspective, sparsity. Prior studies [52, 53] have proven that the health-related frequency bands are much fewer than those related to environments, and [54] has proven that causal features are much sparser than spurious ones. Moreover, some studies in causal representation learning have shown that applying sparsity regularization on the adjacency matrix of features helps prevent the model from overfitting and spurious solutions [55]. Motivated by these, we introduce an entropy-based sparsity regularization on the attention scores to encourage only a few frequency bands with significant values while the rest close to zero. Because, based on the definition of entropy [56], a lower entropy indicates a more concentrated probability distribution, which leads to a concentrated attention along frequency bands here. The sparsity regularization is expressed as:

where $p_a=\text{Softmax}\left(\frac{\left(Z_{cls}{W}_h^Q\right)\bullet {\left({\tilde{Z}}_a{W}_h^K\right)}^T}{\sqrt{d}}\right)$ denotes the attention probability associated with the $a$-th frequency band.

3.5. Training and inference

Beyond the above regularizations, the overall optimization objective is summarized as:

where $\alpha$, $\beta$ and $\gamma$ are the balancing parameters.

The workflow of the proposed method is presented in Table 1.

4.1. Dataset description

We study the performance of the proposed model with 3 datasets as follows, and their key attributes are summarized in Table 2, the test rigs are shown in Fig. 2.

Case Western Reserve University Bearing Dataset [57]. This is a vibration signal dataset that contains vibration signals collected by acceleration sensors from 6205-type ball bearings of 4 individual health states in different severities under 4 working conditions. The 4 health states include normal (N), inner race fault (IR), outer race fault (OR), and ball fault (B). These faults are single-point faults produced through electro-discharge machining with fault diameters of 7 mils, 14 mils, and 21 mils. The vibration signals are collected with a 12 kHz sampling frequency.

Paderborn University Bearing Dataset [58]. This is a vibration signal dataset collected by acceleration sensors from 6203-type ball bearings of 4 health states in different severities under 4 working conditions. We focus on 3 health states: normal (N), inner race fault (IR), and outer race fault (OR). These faults are single-point faults generated through life-accelerated degradation processes (e.g., pitting and indentations) with fault diameters ranging from less than 2 mm to more than 4.5 mm. The vibration signals are collected with a 64 kHz sampling frequency. In our experiments, to preserve enough revolutions for each sample, we downsample the data to 12 kHz.

Harbin Institute of Technology Bearing Dataset [59]. This is a vibration signal dataset that contains vibration signals collected by acceleration sensors from ball bearings of 3 individual health states on a real aeroengine under 25 working conditions. The 3 health states include normal (N), inner race fault (IR), and outer race fault (OR). These faults are produced through wire cutting with a fault depth of 0.5 mm and a fault length of 0.5 mm. The vibration signals are collected with a 25 kHz sampling frequency.

4.2. Implementation details

In our experiments, each sample’s vibration data is first resampled using angular resampling, following [60], to align signals such that each revolution contains an equal number of sampling points within each sample. Specifically, each sample consists of 2048 sampling points, corresponding to 4 revolutions with 512 points per revolution. The Haar wavelet is used as an example. For each sub-signal of frequency band, we set the number of cycle frequency as 128. For the MoE block, we set 6 experts, where each expert consists of 2 MLP layers (256 hidden neurons and a GELU activation) with a learnable similarity matrix of size ${\mathbb{R}}^{256\times 256\times 256}$. For each input feature, the multi-view router selects the TOP-1 expert index out of 6 and routes it to the corresponding one. The multi-head attention mechanism is configured with 8 heads.

Fig. 2Test rigs of each dataset

a) CWRU dataset

b) PU dataset

c) HIT dataset

In the training process, an AdamW optimizer is adopted with a weight decay of 3×10^-4 and an initial learning rate of 1×10^-4, which decays with a cosine annealing schedule. The balancing parameters $\alpha$ and $\beta$ follow a cosine-based growth schedule from 0.01 to 1.0 and from 0.01 to 0.1, respectively, and $\gamma$ is set to be a fixed value of 0.01. For ${\mathcal{L}}_{mono}$, the cycle frequencies are divided into 16 groups, with each consists of 8 adjacent cycle frequencies. Training in the following experiments is performed over 300 iterations with a batch size of 128. All experiments are conducted with Pytorch in Python on an NVIDIA GeForce RTX 2080 Ti GPU.

4.3. Performance comparison experiments

We first test our method in 3 cases with a gradually increasing level of difficulty. In these case studies, we compare our model with the recent representative baseline methods of multi-source domain generalization for bearing fault diagnosis: WhiteningNet [21], DGNIS [16], CDDG [25], ACRLN [4], CIMSDG [27]. In addition, we select an ensemble method using 19 traditional multi-domain features [61] as the basic baseline to evaluate the difficulty of tasks. To ensure the reliability and effectiveness of the experiment results, we conduct experiments for each method five times for each task. The diagnosis performance is indicated with five performance indicators, which are accuracy, fault recall, fault precision, normal recall, and normal precision, defined as:

– Case 1 (Cross-domain transfer on CWRU dataset): With 4 distinct working conditions, we establish 4 different cross-domain tasks. The data are balanced among 4 health states, with 300 samples for each health state in each source domain and 300 samples for each health state in each test domain.

The experimental results of our method, along with the baseline methods, are presented in Table 3 and Fig. 3. Additionally, the comparison results of the detailed performance on each sub-dataset for C2+C3+C4→C1 are shown in Fig. 4. Our model achieves the highest overall average accuracy of 99.33 %. It can be observed that our model and the comparison methods all perform well with only a small performance gap among them. It is because the vibration data in the CWRU dataset has already been denoised and the distribution gap is small in this case, thus, the improvement from adaptive feature extraction with causal localization is not obvious as in the following cases.

Fig. 3Performance comprehensive comparison in Case 1

a) C2+C3+C4→C1

b) C1+C3+C4→C2

c) C1+C2+C4→C3

d) C1+C2+C3→C4

Fig. 4Detailed performance on each sub-dataset of case 1: C2+C3+C4→C1

a) WhiteningNet

b) DGNIS

c) CDDG

d) ACRLN

e) CIMSDG

f) Ours

– Case 2 (Cross-domain transfer from real fault to real fault on PU dataset): With 4 distinct working conditions, we establish 4 different cross-domain tasks. The experimental protocol employs 9 sub-datasets, including 3 normal sub-datasets (K001, K002, K003), 3 inner race fault sub-datasets (KI14, KI16, KI17), and 3 outer race fault sub-datasets (KA04, KA15, KA16). The data are balanced among 3 health states, with 1200 samples for each health state in each source domain and 1200 samples for each health state in each test domain.

The experimental results of our method, along with the baseline methods, are presented in Table 4 and Fig. 5. Additionally, the comparison results of the detailed performance on each sub-dataset for P1+P3+P4→P2 are shown in Fig. 6. Our model achieves high accuracy among all cases and the highest overall average accuracy of 97.79 %. In the most challenging case (P1+P3+P4→P2), our model recognizes better in each sub-dataset and reaches a total accuracy of 94.25 % with a 21.84 % improvement over the best compared method. We also observe that our model has an average accuracy of 72 % on KA15, since the damage type of outer race fault in KA15 (indentations) is different from K04 and K16 (pitting). Although diagnosis under such difference is not the focus of this paper, our model can still handle it to a certain extent.

Fig. 5Performance comprehensive comparison in Case 2

a) P2+P3+P4→P1

b) P1+P3+P4→P2

c) P1+P2+P4→P3

d) P1+P2+P3→P4

Fig. 6Detailed performance on each sub-dataset of case 2: P1+P3+P4→P2

a) WhiteningNet

b) DGNIS

c) CDDG

d) ACRLN

e) CIMSDG

f) Ours

– Case 3 (Cross-domain transfer from artificial fault to artificial fault on HIT dataset): This case is designed to evaluate the transfer performance under large-span working conditions. We establish 4 different cross-domain tasks. The data are balanced among 3 health states, with 270 samples for each health state in each source domain and 270 samples for each health state in each test domain.

The experimental results of our model, along with the baseline methods, are presented in Table 5 and Fig. 7. Additionally, the comparison results of the detailed performance on each sub-dataset for H2+H3+H4→H1 are shown in Fig. 8. Our model achieves high accuracy among all cases and the highest overall average accuracy of 97.95 % among comparison methods. In the most challenging case (H2+H3+H4→H1), our model recognizes outer race fault under working condition of low speed (1500-O) and reaches a total accuracy of 96.79 % with a 6.91 % improvement over the best compared method. This demonstrates that, even in the case of large-span transferring from high-speed condition to low-speed condition, our model can handle the weak fault characteristics and maintain good performance.

Fig. 7Performance comprehensive comparison in Case 3

a) H2+H3+H4→H1

b) H1+H3+H4→H2

c) H1+H2+H4→H3

d) H1+H2+H3→H4

Fig. 8Detailed performance on each sub-dataset of case 3: H2+H3+H4→H1

a) WhiteningNet

b) DGNIS

c) CDDG

d) ACRLN

e) CIMSDG

f) Ours