#### 3.1. Comparison of Ensemble Decision Trees (EDT) Algorithms

Validation of the two proposed AI models are illustrated in

Figure 3 for the training part including 48,051 data (left) and testing part with 20,593 data (right). Error frequency are plotted in function of relative errors between target and output values for EDT Bagged (blue line) and EDT Boosted (orange line) algorithms. In both training and testing parts, the EDT Boosted exhibit more error between output and target, as the values ranging from −1 to 6. The predicted outputs are ranging from 1 time lower to 6 times higher than the target ones. As regard to the training part, the over predicted values outside the range −1 to 1 were 1880 samples (or 3.9%) for EDT Boosted and 275 samples (or 0.57%) for EDT Bagged. For the testing part, the over predicted values outside the range −1 to 1 were 893 samples (or 4.3%) for EDT Boosted and 163 samples (or 0.79%) for EDT Bagged, showing reasonable performance of the AI models. Considering the EDT Bagged algorithm, the error range between output and target is rather narrower than the EDT Boosted model, i.e., ranging from −0.5 to 2. The peaks (blue line) of both training and testing parts are centered at 0, showing excellence prediction capability of EDT Bagged. For both AI algorithms, the number of over predicted values are more than that of under predicted values, confirmed by the error curves are rather on the right side of 0. It is worth noticed that for this simulation, the error measurements are: RMSE = 24.38, MAE = 9.70, R

^{2} = 0.989 for training EDT Bagged, RMSE = 28.28, MAE = 9.91, R

^{2} = 0.986 for testing EDT Bagged, whereas these values are RMSE = 141.48, MAE = 79.64, R

^{2} = 0.852 for training EDT Boosted, RMSE = 144.80, MAE = 81.80, R

^{2} = 0.843 for testing EDT Boosted. In general, both of methods show good predictive capability based on the validation criteria R

^{2}, RMSE and MAE. Comparing the prediction performance using testing parts, EDT Bagged appears a better candidate for this problem, as the RMSE, MAE, and the number of wrong predictions are smaller than those of EDT Boosted, whereas the R

^{2} values are higher than that of the EDT Boosted model.

In validation stage, it is difficult to conclude the performance of given AI algorithms unless a fully analysis is performed. As 70% of data are randomly taken to train the AI prediction tools, the corresponding performance measurements (RMSE, MAE, or R

^{2}) are different for each simulation [

63]. Such values depend of the choice of combination of 70% input data and can be varied from one to another simulation. In order to check the robustness of EDT Bagged and EDT Boosted models, thus, 1000 different arrangements of data were generated using uniform distribution and the error measurements in each case were collected. RMSE, MAE, or R

^{2} errors for 1000 different simulations are plotted under scatter points in

Figure 4 for training and testing parts. The variation of error measurements in the training part is rather smaller than that of the testing part in all cases, which clearly demonstrate the dependence of the predicted results on input combination. In the case of EDT Boosted model (

Figure 4a–c), the training and testing part vary around the same values, i.e., RMSE = 141.21, MAE = 79.65, and R

^{2} = 0.855. Regarding EDT Bagged algorithm (

Figure 4d–f), the training results are RMSE = 25.82, MAE = 9.83 and R

^{2} = 0.988, whereas the testing one are RMSE = 23.78, MAE = 9.14, and R

^{2} = 0.990. This indicates that EDT Bagged outperforms EDT Boosted. However, both EDT Boosted and EDT Bagged algorithms possess ability to well-predict bubble dissolution time in the SLS process.

The normalized statistical convergence analysis of the obtained results is calculated using Equation (3) and plotted in

Figure 5. It is observed that over two Monte Carlo runs for EDT Boosted model are required to achieve the convergence within the 1% range around the converged values. On the contrary, RMSE and MAE criteria require at least 30 simulations to achieve converged values. For R

^{2} criterion, both algorithms are converged in a 1% error range without any simulations. These results demonstrate that the two proposed AI algorithms are promising methods for the prediction of bubble dissolution time with an interesting convergence rate. Out of these, the EDT Bagged is better in term of precision than the EDT Boosted but it requires more simulations to achieve converged statistical values.

#### 3.2. Sensitivity Analysis of Input Parameters

Prediction of bubble dissolution time is a complex problem in which physical and mechanical equations are highly coupled and nonlinear. Thus, the sensitivity analysis is then carried out in order to evaluate the impact of each single input parameter to the predicted output. This could be an attempt to reduce the input space if one input variable is found not affect the final prediction results. All the input parameters were successively excluded from the input space by setting the column to zero values. Monte Carlo simulations were then performed in order to quantify the influence of each parameter, thanks to the prediction performance, i.e., RMSE, MAE, and R

^{2}. In excluding successively I

_{1}, …, I

_{7}, a total number of 7 groups of analysis were generated and simulated with a number of 1000 runs for each group. The statistical performances (RMSE, MAE, and R

^{2}) of two AI models are highlighted in

Table 2. Based on mean values of RMSE, MAE and R

^{2} over 14,000 simulations, the EDT Bagged outperforms the EDT Boosted. Moreover, standard deviation values of EDT Bagged model are smaller than that of the EDT Boosted, indicating that the EDT Bagged algorithm is a more stable prediction technique.

Let us consider the case where input parameters I

_{1} to I

_{7} are successively excluded from the input space. It is observed that when I

_{3} is excluded from the prediction process, highest mean values of RMSE and MAE are obtained, i.e., RMSE = 220.44, MAE = 165.46 for EDT Bagged algorithm and RMSE = 228.18, MAE = 149.14 for EDT Boosted model. Regarding R

^{2}, lowest mean values are obtained: R

^{2} = 0.122 for EDT Bagged algorithm and R

^{2} = 0.168 for EDT Boosted model. This means that without the diffusion coefficient (I

_{3}) in the input space, it is difficult to achieve acceptable performance of the AI prediction tools. Therefore, the diffusion coefficient (I

_{3}) is the most important input variable in the prediction process of bubble dissolution time. Although there is no known analytical solution to predict the bubble dissolution process in Selective Laser Sintering until now, several observations in good agreement with the results can be found in the literature. Kontopoulou et al. [

8], Gogos [

18] reported the kinetic of bubble shrinkage and concluded that such mechanism is mostly controlled by the diffusion of air from the bubble to the polymer melt. The distribution of the resulted dissolution time was highly distributed in the range between 20 s and 200 s. This in good agreement with this finding, as the mean value of I

_{3} was rather high, i.e., 4.1 × 10

^{−9} m

^{2}·s

^{−1}.

Based on the RMSE, MAE, and R

^{2} values, the second and the third important input factors in the prediction process are the initial concentration (saturation rate-I

_{6}) and the initial bubble size (I

_{1}). Indeed, without I

_{6}, the as-obtained errors are: RMSE = 124.63, MAE = 71.64, R

^{2} = 0.719 for EDT Bagged algorithm, whereas RMSE = 184.26, MAE = 101.63, R

^{2} = 0.554 in case of EDT Boosted model. Besides, in excluding I

_{1} out of the input space, the corresponding errors are: RMSE = 100.80, MAE = 48.86, R

^{2} = 0.816 for EDT Bagged algorithm, whereas RMSE = 157.11, MAE = 87.85, R

^{2} = 0.742 for EDT Boosted model. The higher values of RMSE, MAE and the lower values of R

^{2} compared to that of the simulation using 7 inputs indicates the necessity of these variables in the prediction of bubble dissolution time which is in good agreement with the work of Kontopoulou et al. [

8] where the authors mentioned that the air concentration and bubble initial size were found to be of great important in the bubble shrinkage process. Again, the air initial concentration distribution was taken to achieve mean value of I

_{6} = 0.54, SD = 0.37, and the initial bubble size mean value I

_{1} = 112.4 µm, SD = 19.37 µm. The skewness in the input space affected the bubble dissolution time to be in a range between 20 s–200 s, which was in good agreement with observations in the literature.

In our previous work [

17], the contribution lies on the simulation tool for the prediction of bubble dissolution process. The polymer domain is generally considered an infinite domain in the literature, whereas we have introduced the parameter that accounting for the size of the polymer domain (I

_{2}). In the sensitivity analysis of this study, such parameter is found to be more important than the others (I

_{4}, I

_{5} and I

_{7}). The error when excluding I

_{2} are RMSE = 58.84, MAE = 26.57, R

^{2} = 0.938 for EDT Bagged algorithm and RMSE = 142.84, MAE = 80.54, R

^{2} = 0.843 for EDT Boosted model. The surface tension (I

_{4}), applied pressure (I

_{7}), and viscosity (I

_{5}) are found to be the most insensitive input parameters for the prediction problem (

Table 2). The role of these parameters has also been discussed in several works relating the SLS process, such as the surface tension was neglected to derive an analytical solution [

64], or its effect is negligible under certain condition, i.e., small degree of saturation [

18]. The bubble shrinkage process is not influenced considerably by melt viscosity, as reported in the work of Kontopoulou et al. [

8]. Last but not least, the effect of pressure is important only in case when it is applied after the formation of the bubble [

8,

18]. In our developed numerical tool, the pressure is considered constant during the whole SLS process [

17]. Therefore, no conclusion on the effect of pressure can be deduced at this moment.

The histogram of RMSE, MAE, and R

^{2} values of 14,000 simulations are plotted in

Figure 6. All the peaks are rather narrow, demonstrating small values of standard deviation. The three most influenced parameters are easily observed, as they are separated to the rest. However, depending on the AI models, other input parameters can be detected or not. In case of EDT Boosted algorithm, the curves relating I

_{2}, I

_{4}, I

_{5}, and I

_{7} are superimposed for RMSE, MAE and R

^{2} criteria. Regarding the EDT Bagged algorithm, these input parameters can be slightly identified by RMSE and R

^{2} criteria, whereas it is difficult for MAE criterion. As a conclusion, sensitivity analysis requires not only adapted error criteria but also a good prediction model that can capture the difference influence between variables.

The normalized statistical convergence analysis of the results are plotted in

Figure 7 (see also Equation (3)), for the cases of three most influenced input parameters. Again, it is observed that over two Monte Carlo simulations for both EDT Boosted and EDT Bagged models are required to achieve the convergence within the 1% range around the converged values. On the contrary, for R

^{2} criterion, both algorithms are converged in a 1% error range within 10 Monte Carlo simulations. These results demonstrate that the two proposed AI algorithms are potential candidates for the prediction of bubble dissolution time with an interesting convergence rate. Out of these, the EDT Bagged is better in term of precision than the EDT Boosted but it requires more simulations to achieve converged statistical values.