## 3 Proposed Method  
### Problem Formulation  
Generally, this study was conducted under the following assumptions:  
1. Due to the diversity in operating conditions, although the source domain and the target domain are interrelated, they follow the distinct distributions.
2. The purpose of intelligent fault diagnosis in various domains is consistent.
3. The source-domain data (whether labeled or unlabeled) and all unlabeled data from the target domain can be employed to train networks.
4. During training, not only are signals from the target domain unlabeled, but most signals from the source domain are also unlabeled.  
The above assumptions simulate the actual industrial situation where signal labels are insufficient, i.e., a large amount of vibration signals can be collected, but only a small part of them have health condition labels. Existing methods cannot obtain satisfactory diagnosis results.  
Let \(\mathbf{X}\) denote the input signal space and \(\mathbf{Y}=\{1,2,\ldots,N_{c}\}\) represent the set of \(N_{c}\) health conditions, for any sample \((\mathbf{x}_{i},\mathbf{y}_{i})\) with \(\mathbf{x}_{i}\in\mathbf{X},\mathbf{y}_{i}\in\mathbf{Y}\). A partial unlabeled source domain \(\mathcal{D}_{s}\) of \(n_{s}\) samples and an unlabeled target domain \(\mathcal{D}_{t}\) of \(n_{t}\) samples are available. The \(\mathcal{D}_{s}\) and the \(\mathcal{D}_{t}\) can be formulated as  
\[\mathcal{D}_{s}=\{(\mathbf{x}_{i}^{s},\mathbf{y}_{i}^{s}=C_{h})\}_{i=1}^{n_{su}}\cup\{ (\mathbf{x}_{i}^{s},\mathbf{y}_{i}^{s})\}_{i=n_{su}+1}^{n_{sl}} \tag{3}\]  
\[\mathcal{D}_{t}=\left\{\left(\mathbf{x}_{i}^{s},\mathbf{y}_{j}^{s}=0\right)\right\}_{ j=1}^{n_{t}} \tag{4}\]  
where \(\left(\mathbf{x}_{i}^{s},\mathbf{y}_{i}^{s}\right)\) and \(\left(\mathbf{x}_{i}^{t},\mathbf{y}_{i}^{t}\right)\) denote the samples of \((\mathbf{x}_{i},\mathbf{y}_{i})\) in \(\mathcal{D}_{s}\) and \(\mathcal{D}_{t}\), respectively; \(C_{h}\) denotes the labeled machine health conditions; \(n_{su}\) and \(n_{sl}\) mean the number of unlabeled and labeled source domain data, respectively; and \(n_{s}=n_{su}+n_{sl}\). Similarly, \(n_{t}\) denotes the amount of target domain data. \(\mathcal{D}_{s}\) and \(\mathcal{D}_{t}\) are sampled from joint distributions \(P_{s}(\mathbf{X},\mathbf{Y})\) and \(P_{t}(\mathbf{X},\mathbf{Y})\), and \(P_{s}\
eq P_{t}\). Particularly, since the label spaces of the source and target domains are identical in this paper, \(\mathcal{D}_{s}\) and \(\mathcal{D}_{t}\) are considered as being sampled from various marginal distributions--\(P_{s}(\mathbf{X})\) and \(P_{t}(\mathbf{X})\).  
The purpose of this paper is to generate a source domain pseudo-label data set \(D_{s}^{Pseudo}=\left\{\left(\mathbf{x}_{i}^{s},\mathbf{y}_{i}^{s}\in\mathcal{C}_{h} \right)\right\}_{i=1}^{n_{su}}\cup\left\{\left(\mathbf{x}_{i}^{s},\mathbf{y}_{i}^{s} \right)\right\}_{i=n_{su}+1}^{n_{sl}}\) of \(n_{su}\) signals in operating conditions based on \(\mathcal{D}_{s}\). The \(D_{s}^{Pseudo}\) is defined with the following formula:  
\[D_{s}^{Pseudo}=\left\{(\mathbf{x}_{i}^{s},\mathbf{y}_{i}^{s}\
eq C_{h})\right\}_{i=1} ^{n_{su}}\cup\{(\mathbf{x}_{i}^{s},\mathbf{y}_{i}^{s})\}_{i=n_{su}+1}^{n_{sl}} \tag{5}\]  
Afterwards, feature extractors and domain discriminators are constructed for minimizing the feature extractor loss and maximizing the discriminator loss so as to retrieve generalization features and reduce the network loss.