## 2 Problem Statement  
The signal of a working machine was inspected by sensors placed in the specified location. \(x_{0}\), \(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{t}\) is the health indicator inspected from time 0 to \(t\), which is recorded as \(\mathbf{x}_{0:t}\) for convenience. \(r_{0}\), \(r_{1}\), \(r_{2}\), \(\ldots\), \(r_{t}\) is the corresponding RUL from time 0 to \(t\), which is recorded as \(\mathbf{r}_{0:t}\) for convenience. The failure time of the machine is recorded as \(\tau\). The RUL at time \(t\) is recorded as \(r_{t}\)=\(\tau-t\). The LSTM network is typically used in the RUL prediction process in order to learn potential relationships between the input data and the RUL. The LSTM network is assumed as a predictor \(\phi\). The RUL predicted by the predictor at time \(t\) is recorded as \(\alpha_{t}\). All the parameters of the LSTM network are assumed as \(\Theta\). The process of the RUL prediction based on the predictor \(\phi\) can be described as  
\[\phi(\mathbf{x}_{0:t};\Theta)=\alpha_{t}. \tag{1}\]  
However, the lack of run-to-failure data hinders the training process of the LSTM network. Unless the LSTM model \(\phi\) is well-trained and has enough data, it is impossible to build a mapping between health metrics and corresponding RUL. In the actual industry, fault data are limited because the machine always works in a healthy condition most of the time. Thus, run-to-failure data are hard to acquire. Data augmentation is essential for RUL prediction. In this paper, the proposed data augmentation method based on degradation modeling and Sobol sampling are introduced in detail in Section 3.1. Another problem is that the RUL prediction based on the conventional LSTM ignored the relationship between predictions at different times with the same data stream. The uncertainty of the predicted result can not be controlled over the time horizon. The training process of the conventional LSTM network aims to minimize the differences between the real RUL and the predicted RUL at a specific time. The loss function of the conventional LSTM network is  
\[Loss(\mathbf{x}_{0:t},\mathbf{r}_{0:t};\Theta)=\sum_{i=0}^{t}(r_{i}-\phi( \mathbf{x}_{0:i};\Theta))^{2}, \tag{2}\]  
where \(r_{i}\) denotes the real RUL at time \(i\) between time 0 to \(t\), \(x_{i}\) denotes the input signal at time \(i\). In the training process, the optimal parameter \(\Theta^{*}\) of the LSTM can be acquired when minimizing the loss function as  
\[\Theta^{*}=\arg\min_{\Theta}Loss(\mathbf{x}_{0:t},\mathbf{r}_{0:t};\Theta), \tag{3}\]  
the shortcoming of the conventional LSTM prediction is that the predictions at different times are mutually independent. The time-dependence among the RUL prediction sequence is not considered in the conventional LSTM model. Unexpected fluctuation of the predicted RUL may occur because the mutual dependence of the RUL prediction is not concerned according to the loss function described in Equation (2). In order to get a reliable prediction without controllable fluctuations, the uncertainty among the RUL prediction over the time horizon should be taken into consideration. The increment of the predicted RUL \(\alpha_{t}\) and \(\alpha_{s}\) is assumed to follow the normal distribution [35] as  
\[\alpha_{t}-\alpha_{s}\sim\mathcal{N}(s-t,(t-s)/c), \tag{4}\]  
where the \(s-t\) is the mean of the normal distribution. \((t-s)/c\) is the variance of the normal distribution. The parameter \(c\) can be used to control the uncertainty of the variance. This property of the predicted RUL contributes to the assumption of the Wiener process on the predicted RUL [35].