## 1 Introduction
By simultaneously considering time correlation and feature correlation in the condition monitoring data, the method can effectively deal with the time-feature dual correlation graph structure, so as to more accurately and stably predict the amount of time for which the bearings can work safely. However, deep learning performance depends on large amounts of high-quality training data, whereas full-life bearing data is scarce in practical industrial applications, and the model's performance drastically decreases across different operating conditions (load, speed) ([PERSON] et al., 2018). This is because RUL prediction methods based on deep learning commonly assume that the training data (source domain) and the test data (target domain) obey the same distribution. Unfortunately, rolling bearings usually work under various working conditions, which leads to differences in data distribution. Domain adaptation in transfer learning has emerged as a promising approach to address data distribution differences in RUL prediction ([PERSON] et al., 2018). It is usually categorized into two approaches: metric-based learning and adversarial-based learning ([PERSON] et al., 2018). For example, [PERSON] et al. (2018) constructed a number of different models for life prediction of different bearings using the maximum mean difference approach and incorporating bi-directionallong short-term memory networks. [PERSON] et al. [18] combined two models, multi kernel MMD (MK-MMD) and CNN, to innovate a bearing residual life prediction model with migratability. [PERSON] et al. [19] embedded the MMD metric into the constructed bi-directional gated recurrent unit (BiGRU) model. The model is then trained for the phenomenon of differences in data distributions using the ideological approach of transfer learning. [PERSON] et al. [20] devised a multi-scale adversarial domain adaptation framework by integrating deep neural networks with transfer learning paradigms, wherein the Wasserstein distance metric was leveraged to mitigate domain shift challenges in cross-condition remaining useful life prognostics for complex mechanical systems. [PERSON] et al. [21] combined a long short-term memory (LSTM) network with the framework of a domain adversarial neural network (DANN) to propose a LSTM-DANN model. Excellent prediction results have been achieved under different operating conditions and different bearing failure scenarios. Compared to adversarial learning-based approaches, metric learning-based approaches minimize the difference in distribution between source and target domain features as much as possible. However, under harsh and diverse operating conditions, they suffer from a limited ability to learn domain-invariant features, leading to negative transfer due to the risk of poor distribution alignment caused by the inadequate discrepancy representation of MMD's mean statistic in the reproducing kernel Hilbert space (RKHS). Furthermore, normal data lacks degradation information, which significantly increases adaptation difficulty in domain adaptation. To solve the above problems, an enhanced cross-working condition RUL prediction method for rolling bearings, based on an initial degradation detection-enabled joint transfer metric network, is proposed. First, first prediction time (FPT) is applied to identify the change points between the normal and degradation stages, with degradation stage information used as the training target to enhance domain adaptation efficiency and training performance. Secondly, a joint domain adaptation loss is proposed to enable strong domain adaptation by combining distribution discrepancy and adversarial training mechanisms, where representation subspace distance and variance discrepancy representation are applied to the final layer of the mapping regression network to mitigate data distribution discrepancies and ultimately acquire cross-domain invariant features.