## 3 Results and Discussion
### Performance of the Developed ANN Model  
The highly efficient ANN model was developed to predict the thermal behavior of mixed polymers. Specifically, the weight left percent of the tested mixtures, as a function of temperature, heating rate, and mixture composition, was targeted to be predicted by the proposed ANN model. By the \"nntool function\" in MATLAB 2021a, developed by MathWorks Inc., Natick, Massachusetts, USA, the experimental datasets used for the training stage were divided randomly into three sets: 70% for training, 15% for validation, and 15% for testing. In this investigation, an ANN model with a feed-forward and back-propagation scheme was developed using 534 datasets. Then, the developed model was validated using an extra 34 datasets during the simulation stage. The datasets' distribution between training and simulation stages is presented in Table 3.  
The best-developed model has two hidden layers with 15 and 10 neurons, respectively (i.e., 5-15-10-1). In addition, the TANSIG-TANSIG transfer functions and the linear transfer function (Pureline) for the output layer were recommended to be used. Moreover, among different algorithms that were checked, the Levenberg-Marquardt (LM) algorithm was recommended for this network [29]. The network topology of the proposed model is presented in Figure 1.  
Figure 2 presents the performance of the proposed ANN model (5-15-10-1) for all stages. The results indicate an excellent agreement between the experimental (X-axis) and the predicted (Y-axis) values of the output parameter as indicated by the R-value. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test No.** & \begin{tabular}{c} **Training** \\ **Stage No.** \\ \end{tabular} & \begin{tabular}{c} **Simulation** \\ **Stage No.** \\ \end{tabular} & \begin{tabular}{c} **Test No.** \\ \end{tabular} & \
\begin{tabular}{c} **Training** \\ **Stage No.** \\ \end{tabular} &
\begin{tabular}{c} **Simulation** \\ **Stage No.** \\ \end{tabular} \\ \hline
1 & 29 & 2 & 10 & 32 & 2 \\
2 & 32 & 2 & 11 & 32 & 2 \\
3 & 30 & 2 & 12 & 32 & 2 \\
4 & 33 & 2 & 13 & 32 & 2 \\
5 & 33 & 2 & 14 & 33 & 2 \\
6 & 33 & 2 & 15 & 32 & 2 \\
7 & 31 & 2 & 16 & 29 & 2 \\
8 & 33 & 2 & 17 & 25 & 2 \\
9 & 33 & 2 & & 534 & 34 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Datasets distribution between the training and simulation stages for all tests. Figure 1: Topology of the best ANN network (5-15-10-1). The prediction performance of the 5-15-10-1 model was evaluated by the values of R, MAE, RMSE, and MBE. Although a high R-value is preferable, the values of other statistical measurements (MAE, RMSE, and MBE) should be low for a high-efficient ANN model. As presented in Table 4, a high R-value (R \(\geq\) 0.9999), and low values of MAE, RMSE, and MBE were obtained. These values imply the high-efficient performance of the developed model to predict the output parameter. Table 5 shows the parameters of the developed ANN model, and the \"goal\" was the mean square error (MSE), which was set to zero. Figure 3 shows the performance plot through the training, validation, and test errors when predicting the experimental values. The testing was stopped when validation error increased consecutively for six epochs. In this case, an optimum value of MSE (MSE = 0.13715) was obtained. This value is very small and additionally proves the good prediction performance of the developed model [30; 31; 32]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Set**} & \multicolumn{4}{c}{**Training Data Set**} \\ \cline{2-5}  & **R** & **RMSE** & **MAE** & **MBE** \\ \hline Training & 0.99994 & 0.37553 & 0.20154 & \(-\)0.00257 \\ Validation & 0.99992 & 0.37033 & 0.19197 & \(-\)0.04208 \\ Test & 0.99995 & 0.33777 & 0.18669 & 0.021799 \\ All & 0.99994 & 0.36933 & 0.19788 & \(-\)0.00484 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistical parameters of the (5-15-10-1) ANN model. Figure 2: Regression of training, validation, and test plots.