## 1 Introduction  
### Cross-Condition Fault Diagnosis Using Transfer Learning  
Fault diagnosis, the process of identifying faults and pinpointing the type of faults and root cause of the fault when they occur, plays a critical role in the monitoring of machines [1, 2]. With advances in machine learning (ML), data-driven fault diagnosis has recently become a popular topic in academic research and industrial applications [3, 4]. Empowered by the greater data analytical capacity of ML models when compared to traditional methods, data-driven fault diagnosis has demonstrated better diagnostic capacity when dealing with large datasets produced by machines with complex and integrated systems [5]. However, a common assumption in many data-driven fault diagnosis studies is that the data in the diagnosis task follow the same distribution as that used to train the ML models [6]. In reality, this assumption is difficult to achieve, because real machines operate under various operating conditions. For example, different loads and speeds might be imposed on bearings under different operating conditions, and the time-varying wind conditions would place wind turbine gearboxes under constantly varying operating conditions [7]. As a result, ML-based fault diagnosis algorithms would normally face data that have very different distributions from the data used in their initial training, and the distribution discrepancy between the training data and task data degrades the performance of ML-based fault diagnosis [8, 9]. To overcome the performance degradation caused by distribution discrepancy and improve the accuracy of cross-domain fault diagnosis, transfer learning (TL) has been recognised as a promising solution. TL extracts knowledge from a source domain to boost the learning of ML models in a target domain and improve the accuracy of ML models in the target domain [10]. For cross-condition fault diagnosis problems, TL works by leveraging knowledge from a source domain with sufficient, labelled, balanced data with different distributions to improve the performance of ML models on target domain data [11]. There are abundant examples demonstrating the success of applying TL to cross-condition fault diagnosis in the existing literature. Bearings and gearboxes are the most common specific applications for TL-based cross-condition fault diagnosis. Considering the cross-domain fault diagnosis between different loading conditions of bearings, [PERSON] et al. [12] adopted a deep transfer learning method to transfer a convolutional neural network (CNN) trained in the source domain to the target domain data. Using CORrelation ALignment (CORAL) as the distance metric, the fully connected layers of the CNN in the source and target domains are aligned to ensure that the CNN trained by the source domain data under different conditions than the target domain data would perform accurately in the target domain [12]. To minimise the distribution discrepancy between the source and target domain data, [PERSON] et al. [6] used joint distribution alignment (JDA) to align both marginal and conditional distributions so that the classifier trained with source domain data can act accurately on target domain data with different loading conditions. Tests on both bearing and gearbox datasets showed that their proposed TL-based method outperformed the comparison methods in terms of accuracy. Other applications include the cross-condition fault diagnosis for motors by [PERSON] et al. [13]. What is unique about this research is that, apart from considering different invariant working conditions, they included a condition-varying cycle for the test motor under the European Driving Cycle (NEDC). A CNN trained by source domain data was applied to the target domain data, and maximum mean discrepancy (MMD) was used to align the intermediate features to reduce the distribution discrepancy. Their TL-based fault diagnosis method achieved significantly higher accuracy when transferring from invariant working conditions to varying working conditions following NEDC.