## 3 Results and Discussion
### Compression Properties
#### 3.3.2 Regression Models and 3D Plots of Compression Distribution  
Inverse models, found in Equations (7)-(9), were used to express the mechanical properties, which included the fracture strain, yield strength, and compression strength. All of the regression models were dependent on the amount of pressure (P), the H/D ratio, and the wt.% of Al\({}_{2}\)O\({}_{3}\), as well as the interactions among these variables. In the present research, the R\({}^{2}\) and adjusted R\({}^{2}\) had values that were quite near the value of 1. \[ \begin{array}{l}1/\sigma_{y}\,=\,+\,0.005806\,-\,2.63547\,\times\,10^{-6}\, \mathrm{P}\,+\,0.000546\,\mathrm{H/D}\,-\,0.000375\,\mathrm{Al_{2}O_{3}}\,+\,1.141 13\,\times\,10^{-6}\,\mathrm{P}\,\mathrm{*}\,\mathrm{H/D}\,+\,8.67570\,\times \,10^{-8}\,\mathrm{P}\,\times\,\mathrm{Al_{2}O_{3}}\\ -0.000363\,(\mathrm{H/D})^{2}\,+\,0.000014\,(\mathrm{Al_{2}O_{3}})^{2}\\ \end{array}  \tag{7}\]  
\[ \begin{array}{l}1/\sigma_{w}\,=\,+\,0.004246\,-\,8.95796\,\times\,10^{-7}\, \mathrm{P}\,+\,0.000528\,\mathrm{H/D}\,-\,0.000337\,\mathrm{Al_{2}O_{3}}\,-\,5. 56647\,\times\,10^{-7}\,\mathrm{P}\,\times\,\mathrm{H/D}\,+\,2.14145\,\times\,10 ^{-7}\,\mathrm{P}\,\times\,\mathrm{Al_{2}O_{3}}+0.000013\,(\mathrm{H/D})^{2}\,+ \,6.77649\,\times\,10^{-6}\,(\mathrm{Al_{2}O_{3}})^{2}\\ \end{array}  \tag{8}\] \begin{table}
\begin{tabular}{c l c c c c c c} \hline \hline \multicolumn{2}{c}{**Response**} & \multicolumn{1}{c}{**ML**} & \multicolumn{2}{c}{**Training Set**} & \multicolumn{2}{c}{**Testing Set**} \\ \cline{3-8} \multicolumn{2}{c}{} & & **RMSE** & \(\mathbf{R^{2}}\) & **RMSE** & \(\mathbf{R^{2}}\) \\ \hline \multirow{6}{*}{\(\sigma_{y}\) (MPa)} & Linear regression & 8.27 & 0.918 & 5.14 & 0.91 \\  & Regression trees & 3.87 & 0.98 & 6.8 & 0.83 \\  & Random forest regression & 5.69 & 0.96 & 9.01 & 0.71 \\  & Gaussian process regression & 5.63 & 0.96 & 14.44 & 0.80 \\  & Artificial neural networks & 3.17 & 0.98 & 5.19 & 0.99 \\ \hline \multirow{6}{*}{\(\sigma_{\mathrm{uc}}\) (MPa)} & Linear regression & 7.51 & 0.95 & 3.04 & 0.99 \\  & Regression trees & 4.21 & 0.98 & 7.72 & 0.96 \\  & Random forest regression & 5.97 & 0.97 & 12.11 & 0.91 \\  & Gaussian process regression & 3.52 & 0.98 & 4.36 & 0.98 \\  & Artificial neural networks & 3.58 & 0.98 & 4.55 & 0.98 \\ \hline \multirow{6}{*}{\(\varepsilon_{\mathrm{f}}\) (\%)} & Linear regression & 0.59 & 0.87 & 0.68 & 0.91 \\  & Regression trees & 0.24 & 0.97 & 0.51 & 0.86 \\ \cline{1-1}  & Random forest regression & 0.26 & 0.97 & 0.21 & 0.97 \\ \cline{1-1}  & Gaussian process regression & 0.198 & 0.98 & 0.264 & 0.97 \\ \cline{1-1}  & Artificial neural networks & 0.2 & 0.98 & 0.2 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Evaluation metrics for the training and testing groups of compression distribution for ML models. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Input**} & \multicolumn{3}{c}{**Compressive Strength**} & \multicolumn{3}{c}{**Yield Strength**} & \multicolumn{3}{c}{**Fracture Strain**} \\ \hline
**Pressure** & **H/D** & \(\mathbf{Al_{2}O_{3}}\) & **Exp. ** & **Predict** & **Error** & **Exp. ** & **Predict** & **Error** & **Exp. ** & **Predict** & **Error** \\ \hline
800 & 1.5 & 2.5 & 225 & 225.000061 & \(-6.17\times 10^{-5}\) & 315 & 315.14025 & \(-1.40\times 10^{-1}\) & 11.7 & 11.68187 & \(1.81\times 10^{-2}\) \\
600 & 1.5 & 2.5 & 218 & 218.00009 & \(-9.59\times 10^{-6}\) & 287 & 286.98075 & \(1  
(9)  
Figure 16 illustrates the relationship between the mechanical property results for the parameters of the Al-4 Cu/(Al2O3)x composite and its corresponding anticipated values. It is obvious from these data that the suggested regression models were effective since the experimental and forecast values match up well. The following conclusion (Table 8) was drawn from the fact that the majority of the intersections among the experimental and predicted values are located near the median line.