## 4 Experimental Results and Discussion
### First-Order Deposition Parameters  
To investigate the effectiveness of predictive models using neural network (discussed in Section 3.2), several experiments with three deposition parameters were carried out. As mentioned before, these parameters are laser power (P), wire feed rate (F), and travel speed (V), as presented in Table 1. The choices for each of these five sets of parameters result from a test campaign for the development of the process with a wire of 1.2 mm diameter in Inconel(r) 718 [45]. These five sets of parameters frame the stability domain of the process, i.e., they allow stable deposition with different bead morphologies, which is the parameter of interest for our study. Figure 12 shows different bead geometry samples for each parameter variation in the experimental design and for each sample, 10 deposition layers were measured (70% training data, 15% validation data and 15% as testing data). Figure 11: Schematic diagram of bead width and bead height for different deposited layers. Figure 12: Bead geometry samples. In this study, the Levenberg-Marquardt algorithm (LM) was used to study the influence of the deposition parameters (P, F, V) on bead geometries (height and width). The main parameters for LM are the number of hidden layers and nodes. To obtain a good predictive performance a single hidden layer was chosen, with a size of 10 nodes (for more details see e.g., [7]). Additionally, other parameters were also determined by trial and error using 100 iterations for each sample. Figure 13 graphically illustrates a correlation between the measured height and the width predicted by the LM approach for the different deposition parameters. As the figure shows, the points are scattered around the fit line (represents measured bead geometry), which indicates the correlation between the model output values and the actual values. The value of \(R^{2}\) reflects the correlation between the model prediction (\(y^{p}\)) and the corresponding target (\(y^{e}\)). The value of \(R^{2}\) ranges from zero to one, and if \(R^{2}\) is equal to one, it means a perfect correlation, and it can be computed as follows:  
\[R^{2}=1-\frac{\sum_{i=1}^{n}[y_{i}^{p}-y_{i}^{e}]^{2}}{\sum_{i=1}^{n}[y_{i}^{p}- y^{m}]^{2}},\;\;y^{m}=\frac{1}{n}\sum_{i=1}^{n}y_{i}^{p} \tag{10}\]  
where \(n\) is the number of the experimental data. To evaluate the accuracy of the prediction model obtained for each sample, we also computed RMSE, MAE, MAPE. The prediction accuracy of the regression model was judged excellent for RMSE, MAE and MAPE, being close to 0. RMSE is computed to measure the standard deviation of residuals. \[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}^{e}-y_{i}^{p})^{2}} \tag{11}\]  
MAE is the average of the absolute difference between \(y^{e}\) and \(y^{p}\) values in the data-set. \[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}|y_{i}^{e}-y_{i}^{p}| \tag{12}\]  
MAPE is the representation of MAE as a percentage. \[\text{MAPE}=\frac{1}{n}\sum_{i=1}^{n}\frac{|y_{i}^{e}-y_{i}^{p}|}{y_{i}^{e}}*1 00\% \tag{13}\]  
Table 2 compares the bead geometry prediction performance of different samples for the LWAM process. \begin{table}
\begin{tabular}{c c c c} \hline
**Test Trial** & **P (kW)** & **F (m/min)** & **V (m/min)** \\ \hline Sample 1 & 1.6 & 0.6 & 2.0 \\ Sample 2 & 1.6 & 0.3 & 2.2 \\ Sample 3 & 2.0 & 0.9 & 2.2 \\ Sample 4 & 1.6 & 1.2 & 2.2 \\ Sample 5 & 1.4 & 1.4 & 0.45 \\ \hline \end{tabular}
\end{table} 
Table 1: First-order deposition parameters.