# Deformation of Air Bubbles Near a Plunging Jet Using a Machine Learning Approach
## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Model Number**}} & \multirow{2}{*}{**Input Variables**} & \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{\(R^{2}\)} & \multicolumn{2}{c}{**MAE**} & \multicolumn{2}{c}{**RMSE**} \\  & & & & & **[mm]** & **[mm]** & **[mm]** \\ \hline \multirow{8}{*}{\(D_{eq+1}\)} & \multirow{8}{*}{\(1\)} & ARDS & 0.9671 & 0.1132 & 0.1686 & 15.39\% \\  & & Bagging & 0.9749 & 0.0974 & 0.1434 & 13.42\% \\  & & \(R_{b}\), \(V_{b}\), \(R_{b}\), \(F_{Tb}\), & K-Star & 0.9692 & 0.1111 & 0.1582 & 15.27\% \\  & & \(R_{b}\), \(E_{0}\), \(D_{eq0}\) & RF & 0.9734 & 0.1047 & 0.1489 & 14.33\% \\  & & & SVR & 0.9472 & 0.1418 & 0.2020 & 20.18\% \\ \hline \multirow{8}{*}{\(D_{eq+1}\)} & \multirow{8}{*}{\(2\)} & ARDS & 0.9640 & 0.1209 & 0.1748 & 16.31\% \\  & & Bagging & 0.9742 & 0.0973 & 0.1448 & 13.61\% \\  & & \(R_{b}\), \(R_{b}\), \(E_{0}\), \(E_{0}\) & K-Star & 0.9629 & 0.1152 & 0.1717 & 16.29\% \\  & & & RF & 0.9676 & 0.1138 & 0.1645 & 15.42\% \\  & & & SVR & 0.9495 & 0.1466 & 0.2001 & 20.36\% \\ \hline \multirow{8}{*}{\(1\)} & \multirow{8}{*}{\(3\)} & ARDS & 0.8425 & 0.2710 & 0.3552 & 36.80\% \\  & & Bagging & 0.9269 & 0.1706 & 0.2461 & 23.38\% \\  & & \(R_{b}\), \(F_{Tb}\), \(R_{b}\) & K-Star & 0.9214 & 0.1749 & 0.2488 & 24.68\% \\  & & & RF & 0.9512 & 0.1354 & 2.023 & 18.52\% \\  & & & SVR & 0.9546 & 0.1388 & 0.1931 & 19.10\% \\ \hline \multirow{8}{*}{\(D_{eq+2}\)} & \multirow{8}{*}{\(2\)} & ARDS & 0.9536 & 0.1495 & 0.1936 & 20.04\% \\  & & Bagging & 0.9650 & 0.1284 & 0.1675 & 17.35\% \\  & & \(R_{b}\), \(E_{0}\), \(D_{eq0}\) & K-Star & 0.9651 & 0.1278 & 0.1687 & 17.38\% \\  & & & RF & 0.9637 & 0.1346 & 0.1715 & 18.15\% \\  & & & SVR & 0.9413 & 0.1563 & 0.2123 & 22.12\% \\ \hline \multirow{8}{*}{\(D_{eq+2}\)} & \multirow{8}{*}{\(3\)} & ARDS & 0.9499 & 0.1555 & 0.2009 & 20.77\% \\  & & Bagging & 0.9590 & 0.1403 & 0.1820 & 18.88\% \\  & & & K-Star & 0.9481 & 0.1495 & 0.2003 & 20.86\% \\  & & & RF & 0.9553 & 0.1507 & 0.1898 & 20.32\% \\  & & & SVR & 0.9434 & 0.1655 & 0.2144 & 22.39\% \\ \hline \multirow{8}{*}{\(1\)} & \multirow{8}{*}{\(1\)} & ARDS & 0.9261 & 0.1899 & 0.2470 & 25.75\% \\  & & Bagging & 0.9168 & 0.1894 & 0.2610 & 26.04\% \\  & & & K-Star & 0.9170 & 0.1843 & 0.2580 & 25.77\% \\  & & & RF & 0.9386 & 0.1682 & 0.2254 & 22.85\% \\  & & & SVR & 0.9410 & 0.1619 & 0.2147 & 22.58\% \\ \hline \multirow{8}{*}{\(D_{eq+3}\)} & \multirow{8}{*}{\(2\)} & ARDS & 0.9526 & 0.1437 & 0.1980 & 19.29\% \\  & & Bagging & 0.9695 & 0.1181 & 0.1589 & 15.84\% \\  & & \(R_{b}\), \(E_{0}\), \(D_{eq0}\) & RF & 0.9639 & 0.1272 & 0.1705 & 17.28\% \\ \cline{1-1}  & & & RF & 0.9713 & 0.1137 & 0.1541 & 15.38\% \\ \cline{1-1}  & & & SVR & 0.9399 & 0.1555 & 0.2153 & 21.92\% \\ \cline{1-1} \cline{2-1}  & & & ARDS & 0.9451 & 0.1570 & 0.2107 & 21.10\% \\ \cline{1-1}  & & & Bagging & 0.9670 & 0.1204 & 0.1670 & 16.17\% \\ \cline{1-1}  & & & K-Star & 0.9422 & 0.1471 & 0.2094 & 20.86\% \\ \cline{1-1}  & & & RF & 0.9632 & 0.1285 & 0.1734 & 17.39\% \\ \cline{1-1}  & & & SVR & 0.9425 & 0.1645 & 0.2133 & 22.58\% \\ \hline \multirow{8}{*}{\(D_{eq}\)} & \multirow{8}{*}{\(3\)} & ARDS & 0.8932 & 0.2189 & 0.2950 & 29.47\% \\  & & Bagging & 0.8845 & 0.2228 & 0.3036 & 30.51\% \\ \cline{1-1}  & & & K-Star & 0.9031 & 0.1971 & 0.2772 & 27.79\% \\ \cline{1-1}  & & & RF & 0.9170 & 0.1974 & 0.2623 & 26.45\% \\ \cline{1-1}  & & & SVR & 0.9416 & 0.1590 & 0.2162 & 21.74\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Synthesis of the results of equivalent diameter (\(D_{eq}\)) prediction models. the best predictions, while SVR (\(R^{2}\) = 0.9495, MAE = 0.1466 mm, RMSE = 0.2001 mm, RAE = 20.36%) was outperformed by all the other considered ones. RF showed an accuracy very close to that of Bagging, while the ARDS-based variant of Model 2 slightly outperformed the homologous variant of Model 1. Model 3 was generally characterized by a significant decline in forecasting capabilities, however overall performance was still good.