## 5 Conclusions  
In this paper, a bearing RUL prediction method with data augmentation strategy and Wiener-LSTM network with PSO optimization is proposed. The data augmentation method is designed to offer enough training data for the training process. The Wiener-LSTM model is constructed to improve the accuracy of the predicted RUL and control the uncertainty of the result simultaneously. The difference in the predicted RUL result is modeled by the Wiener process to control the uncertainty propagation rate of the result. Wiener process is applied as the connection between the parametric probabilistic model and the LSTM model. The parameters in the Wiener process model and the LSTM network can be updated iteratively by optimizing the joint probability loss function. The proposed Wiener-LSTM method is verified on a popular dataset. The predicted RUL shows better accuracy on test bearings than the normal LSTM model, and the uncertainty of the RUL result can be proved well controlled in the experiments. The stable RUL results with fewer fluctuations predicted by the proposed Wiener-LSTM model are more suitable for decision-making in industrial applications. Two error metrics are applied to verify the validity of the proposed method. The larger MAE error in the testing process indicates a lower accuracy of the model, and conversely, a minor MAE error indicates a higher accuracy. On the test bearing1_3 and bearing1_4, the MAE error of the proposed method is 6.34 and 8.03, which is much lower than that of the traditional LSTM model. The PSO algorithm method is applied here to solve the problem of hyperparameter optimization. An uncertainty propagation modeling framework is proposed for automatic predictors, such as deep learning models. The great potential of the combination of the statistical model and the DL model has been shown in the experiments. Future research on interactive training between the surrogate model and the automatic RUL predictor should be proposed and realized.  
Conceptualization, R.W., L.Y. and Y.D.; methodology, R.W., L.Y., F.Y., R.S. and Y.D.; formal analysis, F.Y. and R.S.; writing--original draft, F.Y., R.W., L.Y., R.S. and J.D.; writing--review and editing, R.W., L.Y., Y.D. and R.S.; supervision, R.W. and L.Y. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Proposed** & **LSTM** \\ \hline Test Bearing & MAE & MAE \\ \hline Bearing1\_3 & **6.34** & 43.6 \\ Bearing1\_4 & **8.03** & 23.8 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Numerical prognostic performance comparisons of different methods.  
Figure 12: The MAE of predicted RUL results by the Wiener–LSTM model and LSTM model. (**a**) The MAE of predicted results on test bearing1_3. (**b**) The MAE of predicted results on test bearing1_4. The blue line is the MAE of results by using the Wiener–LSTM model. The orange line is the MAE of results by using the LSTM model.  
**Funding:** This work was supported by the National Natural Science Foundation of China (Grant No. 51505277, 12074254, 71701143).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The bearings data from the PHM2012 dataset ([[https://hal.archives-ouvertes.fr/hal-00719503](https://hal.archives-ouvertes.fr/hal-00719503)]([https://hal.archives-ouvertes.fr/hal-00719503](https://hal.archives-ouvertes.fr/hal-00719503)), accessed on 22 September 2022).  
**Conflicts of Interest:** The authors declare no conflicts of interest.