# Deformation of Air Bubbles Near a Plunging Jet Using a Machine Learning Approach
## 1 Introduction
However, even if the pressure field is not known, the availability of a good amount of data on the evolution of the size and shape of a suitable number of bubbles, as well as of data on some characteristics of the flow field, leads us to consider an interesting alternative approach for predicting the deformation of individual bubbles: machine learning algorithms. These procedures, particularly suited to dealing with nonlinear regression problems dependent on several variables, have been widely used in recent years in solving a variety of water engineering problems [24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. The complexity of the observed phenomena makes two-phase air-water flows a field of investigation for which machine learning algorithms can represent a very useful tool. However, so far in the literature, there are few studies relating to the application of machine learning algorithms to two-phase air-water flow issues. [PERSON] and [PERSON] [35] developed a novel method to evaluate gas and liquid flow rates in vertical upward gas-liquid pipe flows. This technique consisted of an application of multi-layer backpropagation neural networks on the probability density function and the power spectral density of the normalized output of a differential pressure transducer connected to two axially separated wall pressure taps in the pipe. [PERSON] and [PERSON] [36] used the Regression Tree M5P model, Bagging algorithm and Random Forest algorithm to address some complex problems of wastewater hydraulics, including the air entrainment in a circular drop manhole under supercritical flow. [PERSON] et al. [37] used an adaptive network-based fuzzy inference system (ANFIS) combined with CFD data to predict the macroscopic parameters such as gas velocity in the multiphase reactor. The mentioned studies already show the remarkable potential of approaches based on machine learning algorithms in addressing issues related to multi-phase flows. However, to the best of the authors' knowledge, there are no previous studies based on machine learning algorithms that deal with the evolution of air bubbles in water volume. In this research, three different data-driven models have been developed for the prediction of bubble equivalent diameter (\(D_{eq}\)) and aspect ratio (AR), where AR is equal to the ratio of the minor axis and major axis of the air bubble fitting ellipsoid [38]. Figure 1: Air bubbles entrained by the vertical plunging jet (**a**); individual air bubble path, rising inside the lateral recirculation zone (only the highlighted bubble is at the same time as the recorded frame) (**b**). The models differ in terms of their input variables. Five variants of each model have been built, varying the used machine learning algorithms, which are Additive Regression of Decision Stumps (ARDS), Bootstrap Aggregating (Bagging), K-Star, Random Forest (RF) and Support Vector Regression (SVR). In general terms, the forecasting capability of different machine learning algorithms is strongly dependent on the size and variety of the available dataset. The abovementioned algorithms have been selected because they usually ensure high performance in modelling complex and highly non-linear relationships. The input parameters of the models have been evaluated through experimental measurements carried out using a shadowgraph technique, which will be described in detail in the following section.