## 1 Introduction
In addition, the training of the deep model mainly relies on the data with fault labels to adjust the weight of the model and then find out the mapping function that corresponds the data to the label. But labeling data consumes a lot of labor costs. To tackle this problem and achieve a solution, the concept of transfer learning was developed. The key advantage of transfer learning lies in its ability to leverage labeled data (source domain) from one specific working condition and unlabeled data (target domain) from other, related working conditions for training purposes [21]. It is worth noting that this differs from semi-supervised learning, which only involves a single working condition. Semi-supervised learning trains models by leveraging a combination of a small amount of labeled data and a large volume of unlabeled data, addressing the issue of limited labeled data. In contrast, transfer learning involves source and target domain data from different working conditions. The crux of transfer learning lies in addressing the distributional discrepancy between the two domains. This approach empowers the model to be utilized for fault diagnosis across diverse working conditions. Therefore, transfer learning can build bridges between different fields that follow different probability distributions and establish a learning mechanism that spans different fields [22]. Within the domain of transfer learning, unsupervised domain adaptation (UDA) holds significant importance and has been extensively explored by researchers. It has found successful applications in various fields, including computer vision [23], target detection [24], etc. Since UDA has shown promising results in transfer learning, many researchers have introduced UDA into the field of mechanical fault diagnosis to solve the problem of fault diagnosis under different operating states. UDA can be broadly categorized into two types: base distance and base adversarial [25]. Distance-based unsupervised domain adaptation (UDA) involves mapping features into a shared feature space and employing methods such as Correlation Alignment (CORAL) [26], maximum mean discrepancy (MMD) [27], Central Moment Discrepancy (CMD) [28], and other difference measurement techniques. These methods estimate the distribution gap between the source domain and the target domain, thereby minimizing the distribution discrepancy. The adversarial-based UDA utilizes the interaction between the feature extraction module and the domain discrimination module to achieve the purpose of domain confusion, thereby extracting invariant features from two domains [29]. Inspired by the above methods, many different pieces of research have designed a UDA algorithm for cross-operating condition fault diagnosis. [PERSON] et al. employed CNN for extracting features from rolling bearings. They then minimized the MMD between the multi-layer models and effectively narrowed the distribution distance of feature mappings between the source domain and the target domain [30]. [PERSON] et al. utilized a domain adversarial method based on the Wasserstein distance to train raw signal data, reducing the marginal distribution between the source and target domains in the high-dimensional feature space to extract domain-invariant features [31]. [PERSON] et al. introduced a combination of confrontation and CORAL techniques to tackle the challenge of cross-domain fault diagnosis in gearboxes [32]. However, most of the existing transfer learning methods align marginal distributions (Figure 1a) or conditional distributions (Figure 1b) independently, and they rarely consider both at the same time. To tackle the previously mentioned challenges and improve the effectiveness of cross-domain fault diagnosis, this article propose an innovative approach based on the multi-adversarial joint distribution adaptation network (MAJDAN). The main contributions of this article are as follows:  
1.