## 4 Methodology
### Artificial Neural Network (Modeling and Analysis)
In the back-propagation, the learning rate and the momentum factor are very significant to determine the learning speed and accuracy [57]. The learning rate controls the changes in the weights during the training process. The momentum factor manages the speed of network training. It defines the fraction of preceding weight changes to be included in the current weight changes. Termination of the training depends upon the magnitude of the gradient and the number of validation checks. In case, when the training reaches the minimum of the performance, the gradient becomes very small. For  
Figure 4: ANN model for cutting simulations. example, the training stops when the magnitude of the gradient is less than \(1\times 10^{-5}\). This limit can be adjusted by setting the parameter net.trainParam.min_grad. Figure 5a shows the performance, training statistics, and the convergence plots for testing and training networks of tool-chip interface temperature. The performance plot (mean square error of all data sets) is shown on a logarithmic scale. The training mean square error must show a decreasing trend. Here, the training plot shows a perfect training. The other two plots explain the network simulation results after the training. The training was terminated when the validation error increased to 16 epochs, and the best validation performance was obtained as 16.58 at 10 th epoch. The test set error and the validations set error have also shown the similar characteristics. Figure 5b shows the coefficients of regression. The R-plots explain the significance between the target (desired output) and the ANN output (actual output). The dashed line in each plot represents the targeted values (the difference between the perfect result and outputs). The best-fit linear regression line between the outputs and targets is represented by a solid line. The correlation coefficient (\(R\)) gives the relationship between the outputs and the targets. The maximum value of the correlation coefficient (\(R^{2}\)) and a minimum value of the root mean square error defines a good ANN model. For an exact linear relationship, \(R\) must be closer or equal to one. The values of coefficients for training and testing data are founded to be 0.996 and 0.953, respectively. For a perfect fit, the distribution of data should be along a 45\({}^{\circ}\) line, which shows that the network outputs are equal to the targets. From Figure 5b, it was observed that the targeted output \(R\) for training is 0.99695, validation is 0.98939), and testing is 0.95314. The corresponding total response is 0.999018. \(R=0.999018\) verifies that the ANN output perfectly matches with the target (precise linear relevance). The overall response verifies that the training has produced the optimal results, and the model can entertain the new inputs. Values of all the coefficients are very close to 1, so the training value is highly acceptable. Figure 5d is a caption window that shows the validation of ANN model and also explains the output of the neural network during the training and training developments. Figure 5c shows the model evolution, validation, and the corresponding gradient of epochs. For the temperature prediction model, the gradient of epochs was attained as 31.548. The second ANN model for cutting reaction force was developed by keeping the same input parameters, as shown in Figure 4. The ANN parameters for training and testing are presented in Table 4. The training and performance evaluation of the network was accomplished with 64 data sets, given in Table 3. The standard feedforward backpropagation neural network was considered with Log-Sigmoid transfer function (LOGSIG) and Levenberg-Marquardt algorithm. The optimized network was detected with 10 neurons in the hidden layers. The ANN predicted outcomes were evaluated by comparing with the published experimental results [42]. The ANN predicted values and the performance curves of cutting reaction force are shown in Figure 6a-d. Figure 6a illustrates the