## 4 Modeling and Optimization Concerns
### BPNN Algorithm  
A conventional BPNN algorithm [26], displayed in Figure 11, was applied to predict the machine performance indices. Several methodologies, such as [16; 18], were tested, but the BPNN was the most accurate for predicting the information index and could be applied later in the optimization process with less computational burden. The architecture contained five inputs, three output elements, 20 hidden layers, and one output layer.  
The inputs were the interpolation parameters listed in Table 2, and the outputs were the performance indices, namely, the MCoE, MT, and CVib. The training dataset was from the 1024 experiments described in Section 3, and the validation dataset comprised 64 randomly selected experiments and the parameter ranges in Table 2. Besides, 1024 experiments were conducted for the training dataset; another 64 experiments were conducted for the validation of the training model. The other four experiments were executed for testing the optimization process. A total of 1092 experiments were conducted in the study. In Table 3, the training and validation results for the root mean square error (RMSE) and maximum error (ME) are listed to demonstrate the accuracy of the trained model.  
In the validation dataset, the RMSE and ME of the MCoE were 1.0976 and 5.5356 \(\mu\)m, respectively. The RMSE and ME of the MT were 0.1305 and 0.5847 s, respectively. The RMSE error was approximately 1% when a mean value of MT \(\mu\) equal to 12.69 s was applied. Finally, the RMSE and ME of the CVib were 4.48 and 8.28 mm/s\({}^{2}\), respectively. The correlation coefficients of the predicted values and experimental results were calculated using the Excel CORREL function. The equation for the correlation coefficient for a two random variable array of X and Y is defined as  
\[\text{Correl}(\text{X},\text{Y})=\frac{\sum_{i=1}^{n}(x-\overline{x})(y- \overline{y})}{\sqrt{\sum_{i=1}^{n}(x-\overline{x})^{2}\sum_{i=1}^{n}(y- \overline{y})^{2}}} \tag{1}\]  
Figure 11: Architecture of the backpropagation neural network (BPNN).  
where \(\overline{x}\) and \(\overline{y}\) are the means of the data of x and y, respectively. The correlation coefficients of the predicted values and experimental results for the MCoE, MT, and CVib based on the validation data were 0.9984, 0.9998, and 0.9354, respectively. As indicated in the introduction, the vibration of a machine tool is extremely difficult to predict using the finite element method. The high correlation coefficients suggest the developed model is accurate for designing the interpolation parameters.  
Experimental design is a crucial concern in developing data-driven models. The TED has the advantage of requiring a relatively low number of experiments, which reduces the data acquisition time. However, the results might not be as accurate as those generated through a FFD [27]. To compare the performance of the two experimental design approaches, L16(4\({}^{5}\)) arrays--including five factors and four levels--were used in the TED in this study. Only 16 experiments were conducted with the training dataset. The RMSE and ME of the MCoE were 19.33 and 40.47 \(\upmu\)m, respectively. The RMSE and ME of the MT were 2.1144 and 7.9993 s, respectively. The CVib was 13.52 and 26.12 mm/s\({}^{2}\), respectively. Thus, the accuracy of the TED-based model was much lower than that of the FFD. Although the number of experiments needed in the FFD was much higher than that in the TED, the AAS provided a convenient platform for conducting more experiments with less effort.