# Deformation of Air Bubbles Near a Plunging Jet Using a Machine Learning Approach
## 3 Results and Discussion
Unlike previous models, the SVR-based variant (\(R^{2}\) = 0.9546, MAE = 0.1388 mm, RMSE = 0.1931 mm, RAE = 19.10%) outperformed all the others. ARDS led to significantly less accurate predictions (\(R^{2}\) = 0.8425, MAE = 0.2710 mm, RMSE = 0.3552 mm, RAE = 36.80%), while RF ensured predictions comparable to those of SVR, outperforming both Bagging and K-Star. As for \(D_{\alpha q+2}\), Model 1 led to slightly less accurate results than those relating to the prediction of \(D_{\alpha q+1}\). Again, the Bagging-based variant of Model 1 (\(R^{2}\) = 0.9650, MAE = 0.1284 mm, RMSE = 0.1675 mm, RAE = 17.35%) led to the most accurate predictions. On the other hand, SVR was, again, the worst performing algorithm (\(R^{2}\) = 0.9413, MAE = 0.1563 mm, RMSE = 0.2123 mm, RAE = 22.12%). RF and K-Star showed a comparable performance and both slightly outperformed ARDS. Again, Model 2 predictive capabilities were just below those of Model 1 and Bagging algorithm led to the most accurate forecasts (\(R^{2}\) = 0.9590, MAE = 0.1403 mm, RMSE = 0.1820 mm, RAE = 18.88%), whereas SVR (\(R^{2}\) = 0.9434, MAE = 0.1655 mm, RMSE = 0.2144 mm, RAE = 22.39%) provided the least accurate outcomes. RF, ARDS and K-Star were characterized by their comparable forecasting capabilities in this case. Model 3 was again characterized by a remarkable decrease in accuracy of forecasts, compared to Model 1 and Model 2, with a manifest increase in errors. As in the case of \(D_{\alpha q+1}\), the SVR-based variant (\(R^{2}\) = 0.9410, MAE = 0.1619 mm, RMSE = 0.2147 mm, RAE = 22.58%) led to the best results, while the Bagging based variant (\(R^{2}\) = 0.9168, MAE = 0.1894 mm, RMSE = 0.2610 mm, RAE = 26.04%) provided the worst performance. RF outperformed both ARDS and Bagging. Moreover, the ARDS-based variant outperformed the homologous variant developed to forecast \(D_{\alpha q+1}\). Regarding \(D_{\alpha q+3}\), the forecasting capabilities of Model 1 were in line with those of the homologous forecasting model of \(D_{\alpha q+2}\). In this case RF led to the best forecasts (\(R^{2}\) = 0.9713, MAE = 0.1137 mm, RMSE = 0.1541 mm, RAE = 15.38%), while ARDS and SVR provided the least accurate results. Once again, the predictions of Model 2 were slightly less accurate than those of Model 1. The variant based on the Bagging algorithm (\(R^{2}\) = 0.670, MAE = 0.1204 mm, RMSE = 0.1670 mm, RAE = 16.17%) proved to be the best performing, while those based on ARDS and SVR also in this case led to the least accurate results. Model 3 confirmed itself as the worst performing model. Among the various variants of Model 3, the one based on SVR (\(R^{2}\) = 0.9416, MAE = 0.1590 mm, RMSE = 0.2162 mm, RAE = 21.74%) confirmed to be the most accurate, significantly outperforming all the others. A comparative analysis of the three considered models, in all variants, showed that they do not undergo a significant reduction in forecasting capability of the equivalent diameter of the bubble, gradually passing from \(D_{\alpha q+1}\) to \(D_{\alpha q+2}\) and \(D_{\alpha q+3}\). Therefore, with a good experimental dataset available, an approach based on machine learning may allow us to predict, with good accuracy, the variation in the bubble volume over time. Model 1 generally leads to the best predictions, but it requires a greater number of input variables and, in particular, it needs to know the initial depth of the bubble. Therefore, for practical purposes it may be better to use Model 2, whose high forecasting capabilities demonstrate that, to predict the volumetric deformation of the bubble with satisfactory accuracy, it is enough to know the initial values of the Weber, Reynolds and Eotvos numbers. While it was established that, within Model 1 and Model 2, all the considered algorithms are able to provide good predictions, the variants based on the Bagging algorithm led to the best results, while those based on SVR provided the least accurate results. The latter algorithm, on the other hand, is the one that provided the best results within Model 3.