###### Abstract  
The remaining useful life (RUL) of bearings based on deep learning methods has been increasingly used. However, there are still two obstacles in deep learning RUL prediction: (1) the training process of the deep learning model requires enough data, but run-to-failure data are limited in the actual industry; (2) the mutual dependence between RUL predictions at different time instants are commonly ignored in existing RUL prediction methods. To overcome these problems, a RUL prediction method combining the data augmentation strategy and Wiener-LSTM network is proposed. First, the Sobol sampling strategy is implemented to augment run-to-failure data based on the degradation model. Then, the Wiener-LSTM model is developed for the RUL prediction of bearings. Different from the existing LSTM-based bearing RUL methods, the Wiener-LSTM model utilizes the Wiener process to represent the mutual dependence between the predicted RUL results at different time instants and embeds the Wiener process into the LSTM to control the uncertainty of the result. A joint optimization strategy is applied in the construction of the loss function. The efficacy and superiority of the proposed method are verified on a rolling bearing dataset obtained from the PRONOSTIA platform. Compared with the conventional bearing RUL prediction methods, the proposed method can effectively augment the bearing run-to-failure data and, thus, improve the prediction results. Meanwhile, fluctuations of the bearing RUL prediction result are significantly suppressed by the proposed method, and the prediction errors of the proposed method are much lower than other comparative methods. remaining useful life prediction; bearing; long short-term memory (LSTM); sequence-wise uncertainty control; Wiener process +
Footnote †: journal: Applied Sciences  
maximization algorithm [12], recursive filtering algorithm [13], and so on. The Gamma process and the Inverse Gaussian process, however, are only appropriate for describing monotonous degradation processes. Because of self-healing and small machine repairs, non-monotonous processes are more common in actual industrial settings. The Wiener process is more popular because it can be used both in the monotonous and non-monotonous degradation process. Although statistical model-based methods have implemented several well-established studies in RUL prediction, statistical model-based methods are often developed on a case-specific basis, with the limitation of requiring explicit prior knowledge [14; 15]. The AI-based methods can deal with a large amount of monitoring data by machine learning (ML) techniques without much prior knowledge. Traditional machine learning techniques, such as Support Vector Machine (SVM) [16; 17] and Relevance Vector Machine (RVM) [18], have been widely used in RUL prediction. Compared with the traditional machine learning techniques, deep learning methods show greater potential in dealing with high non-linearity and data complexity. There are some commonly used deep learning (DL) models, such as Auto-Encoder (AE) [19], Convolutional Neural Network (CNN) [20; 21; 22; 23; 24], Recurrent Neural Network (RNN) [25], and so on. [PERSON] et al. [26] used CNN in bearing RUL prediction. Raw signals are transformed into the time-frequency domain features and then mapped with the RUL by the 2D-CNN model. The Long Short-Term Memory Neural Network (LSTM) network is one of the RNN variations that can address the issue of gradient disappearance and gradient explosion. [PERSON] et al. [27] proposed an LSTM with attention mechanism for gear RUL prediction, calculating the contribution of the input during the RUL prediction by using the attention mechanism to improve the performance of the network. However, the AI-based methods still have two limitations: (1) the training process of AI-based methods requires enough data; Thus, lacking run-to-failure data limits the training process. (2) The RUL prediction by the existing deep learning model at different times is mutually independent.