# 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{2}{c}{**Model**} & \multirow{2}{*}{**Input Variables**} & \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{**\(R^{2}\)**} & **MAE** & **RMSE** & **MAE** \\ \hline \multirow{8}{*}{**AR\({}_{+1}\)**} & \multirow{8}{*}{1} & ARDS & 0.5376 & 0.0244 & 0.0317 & 67.89\% \\  & & \(h_{b}\), \(V_{b}\), \(We_{b}\), \(Fr_{b}\), & Bagging & 0.4651 & 0.0254 & 0.0342 & 69.85\% \\  & & \(Re_{b}\), \(Eo\), \(Do_{eq0}\) & K-Star & 0.3505 & 0.0277 & 0.0372 & 76.64\% \\  & & & RF & 0.4002 & 0.0276 & 0.0384 & 72.71\% \\  & & & SVR & 0.2976 & 0.0272 & 0.0373 & 76.96\% \\ \hline \multirow{8}{*}{**AR\({}_{+1}\)**} & \multirow{8}{*}{2} & ARDS & 0.4795 & 0.0283 & 0.0372 & 72.86\% \\  & & & Bagging & 0.3884 & 0.0286 & 0.0376 & 77.28\% \\  & & & K-Star & 0.3573 & 0.0277 & 0.0369 & 77.11\% \\  & & & RF & 0.3302 & 0.0303 & 0.0407 & 79.89\% \\  & & & SVR & 0.3637 & 0.0279 & 0.0374 & 76.38\% \\ \hline \multirow{8}{*}{**AR\({}_{+1}\)**} & \multirow{8}{*}{3} & ARDS & 0.1463 & 0.0400 & 0.0540 & 107.30\% \\  & & & Bagging & 0.1957 & 0.0321 & 0.0429 & 85.76\% \\  & & & K-Star & 0.2257 & 0.0311 & 0.0417 & 83.70\% \\  & & & RF & 0.2038 & 0.0303 & 0.0406 & 84.40\% \\  & & & SVR & 0.3023 & 0.0287 & 0.0387 & 79.55\% \\ \hline \multirow{8}{*}{**AR\({}_{+2}\)**} & \multirow{8}{*}{1} & ARDS & 0.4628 & 0.0284 & 0.0385 & 72.66\% \\  & & \(h_{b}\), \(V_{b}\), \(We_{b}\), \(Fr_{b}\), & Bagging & 0.3401 & 0.0294 & 0.0386 & 80.03\% \\  & & \(Re_{b}\), \(Eo\), \(Do_{eq0}\) & K-Star & 0.3783 & 0.0308 & 0.0420 & 77.99\% \\  & & & RF & 0.3772 & 0.0300 & 0.0409 & 78.33\% \\  & & & SVR & 0.4035 & 0.0280 & 0.0371 & 75.81\% \\ \hline \multirow{8}{*}{**AR\({}_{+2}\)**} & \multirow{8}{*}{2} & ARDS & 0.3354 & 0.0332 & 0.0439 & 83.01\% \\  & & & Bagging & 0.3659 & 0.0312 & 0.0404 & 80.95\% \\  & & & K-Star & 0.4459 & 0.0315 & 0.0411 & 77.78\% \\  & & & RF & 0.4416 & 0.0311 & 0.0417 & 76.21\% \\  & & & SVR & 0.4242 & 0.0332 & 0.0432 & 80.22\% \\ \hline \multirow{8}{*}{**AR\({}_{+2}\)**} & \multirow{8}{*}{2} & ARDS & 0.0557 & 0.0434 & 0.0547 & 105.91\% \\  & & & Bagging & 0.3282 & 0.0349 & 0.0434 & 86.55\% \\  & & & K-Star & 0.4226 & 0.0328 & 0.0411 & 80.80\% \\  & & & RF & 0.3466 & 0.0344 & 0.0448 & 84.39\% \\  & & & SVR & 0.4538 & 0.0334 & 0.0422 & 80.08\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Synthesis of the results of aspect ratio (AR) prediction models. Figure 9: Comparison among equivalent diameters by means of time series; (**a**) Model 1—bubble 1; (**b**) Model 2—bubble 12. With regard to AR\({}_{+1}\), Model 1 showed again the best prediction capability and the ARDS algorithm was the most accurate one (\(R^{2}=0.5376\), MAE = 0.0244, RMSE = 0.0317, RAE = 67.89%), while all the other considered algorithms showed comparable performances. Model 2 showed an appreciable deterioration in forecasting capabilities compared to Model 1, except for the SVR-based variant. The results of Model 3 were even less accurate, especially those of the ARDS-based variant. As for AR\({}_{+2}\), Model 1 outcomes were comparable to the results of the homologous predictive model of AR\({}_{+1}\), in some cases even slightly better. Model 2 predictions also had an accuracy similar to that of Model 1. In particular, from the comparison with Model 1, by analyzing the results of the different Model 2 variants, it could be observed that ARDS was characterized by the deterioration of the predictive capabilities, while K-Star showed an improvement. For the other algorithms, the different metrics led to conflicting results. Model 3 proved to be the less accurate once again, except for the SVR-based variant, which outperformed the analogous variant of Model 2. As for AR\({}_{+3}\), Model 2 outperformed Model 1 in all variants, while Model 3 led to the least accurate predictions, except for the SVR-based variant, the results of which are comparable to the analogous variant of Model 2. Figure 10 shows a comparison between the  
It is evident that the results which most negatively affect the metrics refer to AR \(<\) 0.3.