## 4 Methodology
### Artificial Neural Network (Modeling and Analysis)
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Function** & **Value** & **Description** \\ \hline Training function & - & TRAINLM \\ Adaptation learning function & - & LEARNGDM \\ Network type & - & Feed-forward backprop \\ Transfer function & - & LOGSIG \\ net.trainParam.epochs & 1000 & Maximum number of epochs to train \\ net.trainParam.goal & 0 & Performance goal \\ net.trainParam.showCommandLine & false & Generate command-line output \\ net.trainParam.showWindow & true & Show training GUI \\ net.trainParam.lr & 0.01 & Learning rate \\ net.trainParam.max\_fail & 6 & Maximum validation failures \\ net.trainParam.min\_grad & \(1\times 10^{-7}\) & Minimum performance gradient \\ net.trainParam.show & 25 & Epochs between displays \\ net.trainParam.time & inf & Maximum time to train in seconds \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ANN network and training parameters. performance, training statistics, and the convergence plots for training the network. The performance curve shown in Figure 5(a) demonstrates a perfect training; the test set error, and the validation set error plots also interpret the similar characteristics. Training was completed when the validation error increased to 16 epochs, and the best validation performance was attained as 57.4402 at 10 th epoch. The regression analysis is shown in Figure 5(b). The coefficients of regression value for training and testing data were attained as 0.99801 and 0.96738, respectively. The targeted output value of the validation and corresponding total response are 0.99512 and 0.98768, respectively. The overall results demonstrate a precise linear relevance and validate the model for accommodating the new inputs and simulating the final results. Figure 4(c) illustrates the model validation and the corresponding gradient of epochs. The gradient of epochs was attained as 97.3974. The network configuration, the optimization scheme, and the validation of ANN model are shown as a caption window in Figure 5(d). Figure 5: _Cont._  
A comparison of the numerical simulation results with the ANN predicted values is presented in Table 5. The term \(RF\_Pd\) gives the predicted reaction force, \(Err\) is the error, % \(Err\) is the percent error, \(T\) is the chip-tool interface temperature, and \(T\_Pd\) is the chip-tool interface predicted temperature. The performance of the ANN model elaborates the deviation (error) between the actual and predicted values. In Table 5, the maximum percentage errors of the cutting reaction force and the chip-tool interface temperature are found to be 2.55 and 3.34, respectively. The calculated errors are found reasonable and demonstrate the conformance of ANN predicted results. Figure 5: (**a**) Neural network performance plot for tool-chip interface temperature (\(T\)); (**b**) Neural network regression plots for chip-tool interface temperature (\(T\)); (**c**) Neural network performance and gradient epochs (chip-tool interface temperature); (**d**) Neural training and the output window for chip-tool interface temperature (\(T\)). Figure 6: _Cont._  
The ANN predicted values of tool-chip interface temperatures and reaction forces were compared with the experimental results [42; 58]. Values are presented in Table 6, where the term % \(Err\) shows the percent error; ANN_Sim RF is the ANN simulated reaction force; Exp_RF is the experimental value of the reaction force; ANN_Sim T is the ANN simulated temperature; and Abq Sim_T is the Abaqus simulated chip-tool interface temperature. The ANN simulated values of cutting reaction force and tool-chip interface temperatures are found in good approximation with the experimental results [42]. This shows that the ANN models developed for the temperature and reaction force can be used effectively for predicting the optimal parameters. The maximum percentage errors for ANN predicted reaction force and the chip-tool interface temperatures are 3.4 and 3.2, respectively. Figure 7 implies a good agreement between experimental and ANN simulated outcomes.