## 4 Experiment
### Results and Discussions
#### 4.4.2 Comparison 2  
To verify the effectiveness of the Wiener-LSTM model, the comparison experiments between the Wiener-LSTM model and the LSTM model are taken. In the training process of the LSTM model, the loss function MSE is shown in Figure 10. The training loss function of normal MSE declines rapidly from the beginning to the 10 th epoch and shows a steady trend from the 10 th epoch to the 80 th epoch. The results of the loss function curve show that each model has reached convergence at the 80 th epoch. The same number of epoch should be ensured for the two models for comparison. Thus the training epoch is set for 80. Predicted RUL results by Wiener-LSTM and LSTM models are shown in Figure 11. The blue line in Figure 11a,b represents the real RUL of the bearing1_3 and bearing1_4. The orange line is the result predicted by the Wiener-LSTM model with the loss function in Equation (23). The yellow line indicates the result predicted by the normal LSTM using the MSE loss function. Both models have the same structure: the same layers and the same number of neurons in each layer. Predicted results of the test bearing1_3 are shown in Figure 11a. As for the test bearing1_3, the curve of the predicted RUL based on the Wiener-LSTM model is closer to the real RUL than the results based on the normal LSTM model at the beginning 0-400 s. After 400 s, the predicted RUL based on the Wiener-LSTM model is farther from the real RUL than the results based on the normal LSTM model. Although the Wiener-LSTM model has worse accuracy than the normal LSTM model at the end of the time in Figure 11. The predicted result based on the Wiener-LSTM model has a stable result without large fluctuation. On the contrary, the unstable RUL result with fluctuations is predicted by the LSTM model. The large fluctuation of the predicted RUL would affect the decision-making of the maintain. However, the predicted RUL results by the Wiener-LSTM model are smooth. The assumed Wiener process is proved that can be well fitted by the LSTM model to control the uncertainty of the predicted RUL. The uncertainty of the RUL between continuous time can be controlled by the Wiener-LSTM model. The predicted result based on Wiener-LSTM shows better accuracy than the LSTM model for test bearing1_4 as shown in Figure 11b. The predicted RUL curve is closer to the real RUL (the blue line) than the LSTM model all the time. The curve of the RUL result predicted by the Wiener-LSTM model is smoother than the RUL result predicted by the LSTM model. Fluctuations are obvious in the result of the LSTM model. The uncertainty of the result can be well controlled no matter which test bearing. The reliability of maintenance decisions is improved by the proposed Wiener-LSTM model. Figure 10: The loss function curve of the LSTM model training process. Two stages can be divided among the epoch of the loss function, which include the rapid decline stage and the convergence stage. A normal indicators Mean Absolute Error (MAE) is implemented in this paper to assess the predicted result quantitatively. The functions of the MAE is  
\[MAE=\sum_{i=1}^{N}\left|r_{i}-\alpha_{i}\right|/N, \tag{24}\]  
where the \(r_{i}\) denotes the real RUL at time \(i\), the \(\alpha_{i}\) denotes the predicted RUL at time \(i\). \(N\) is the number of the samples. The MAE of the predicted results by Wiener-LSTM and LSTM models are shown shown in Figure 12. The MAE of the predicted results of the test bearing1_3 is shown in Figure 12a. The MAE of the predicted results of the test bearing1_4 is shown in Figure 12b. The blue line is the MAE result by using the Wiener-LSTM model. The orange line is the MAE result by using the LSTM model. In Figure 12a, The MAE of the Wiener-LSTM model is lower than the LSTM model on the test bearing1_3 at the beginning time 0-600 s. Although the MAE of the LSTM model is mostly the same as the Wiener-LSTM model. The MAE of the LSTM model has large fluctuation. On the contrary, the MAE of the Wiener-LSTM model is smooth all the time.