## 3 Simulation Results and Discussions
### Verification Results  
Table 1 below tabulates the best simulation results obtained with the network settings mentioned in the previous section, for all different datasets (10,000, 20,000, 40,000, and 100,000 data samples). Specifically, the loss function (MAE) results for the \(x,y,z\) electric dipole position and the \(p_{x},p_{y},p_{z}\) electric moments for corresponding test datasets (10,000, 20,000, 40,000, and 100,000 data samples) and the learning rate \(LR=0.01\) is presented. Figure 4: Flow chart of the overall methodology of the present work. Table 1 indicates that the best overall results were obtained when the 40,000 samples dataset was applied. As mentioned in Section 2.2.2, the electric dipole position variables x, y, and z were bounded in the intervals [\(-\)0.25, 0.25] m, [\(-\)0.25, 0.25] m, and [0, 0.5] m, respectively, and thus the reached Mean Absolute Errors of 0.0016 m, 0.0017 m, and 0.0016 m presented in Table 1 for the 40,000 data samples case constitute acceptable and satisfactory position errors. Correspondingly, in the case of the electric moments p\({}_{\rm x}\), p\({}_{\rm y}\), and p\({}_{\rm z}\), which are all bounded in the interval [\(-\)5\(\cdot\)\(e^{-14}\), 5\(\cdot\)\(e^{-14}\)] C\(\cdot\)m, the achieved Mean Absolute Errors are 2.0314\(\cdot\)10\({}^{-16}\)C\(\cdot\)m, 2.1723\(\cdot\)10\({}^{-16}\)C\(\cdot\)m and 2.0643\(\cdot\)10\({}^{-16}\)C\(\cdot\)m for each moment parameter, which are also satisfactory training outcomes. Table 2 presents the best results achieved in the case of 40,000 data samples for various numbers of hidden layers and neurons at each layer. As can be observed, the MAE criterion for all the variables was reduced when four and five hidden layers were applied to the neural network structure, and the optimal solution was given for four hidden layers. This way, the minimization of MAE yielded the optimal configuration. Table 3 presents the results obtained by applying during the training procedure the learning rates \(LR=0.01\), \(LR=0.001\) in the optimal neural network architecture of 4 layers and 580, 280, 280, and 80 neurons at each layer, respectively, for the 40,000 data case. The learning rate \(LR=0.01\) contributed to obtaining better overall results. The test relative error of the electric field vector at the 14 measurement points and for the case of 40,000 data (4000 test data) and learning rate \(LR=0.01\) is depicted in Figure 5. From this Figure, it is readily apparent that the electric field vector can be accurately reproduced from the EDMnet's predicted values (\(x,y,z,p_{x},p_{y},p_{z}\)), as the relative error between the generated and predicted values at the 13 from the 14 measurement points is less than 2%. \begin{table}
\begin{tabular}{c c c} \hline
**Data Sample Size** & **Test** & **Test** \\
**MAE\({}_{\rm x}\) / MAE\({}_{\rm y}\) / MAE\({}_{\rm z}\)** & **MAE\({}_{\rm px}\) / MAE\({}_{\rm pg}\) / MAE\({}_{\rm pg}\)** \\
**(m)** & **(\(\times\)10\({}^{-16}\)C\(\cdot\)m)** \\ \hline
10,000 & 0.0024 / 0.0023 / 0.0026 & 2.6177 / 2.9975 / 2.8126 \\
20,000 & 0.0016 / 0.0017 / 0.0017 & 2.2935 / 2.2030 / 2.2338 \\
40,000 & 0.0016 / 0.0017 / 0.0016 & 2.0314 / 2.1723 / 2.0643 \\
100,000 & 0.0016 / 0.0016 / 0.0016 & 2.2639 / 2.6554 / 2.2791 \\ \hline \end{tabular}
\end{table} 
Table 1: Test Mean Absolute Error for the different datasets-learning rate \(LR=0.01\). \begin{table}
\begin{tabular}{c c c} \hline
**[LI L2 L3 L4 L5]** & **MAE\({}_{\rm x}\) / MAE\({}_{\rm y}\) / MAE\({}_{\rm z}\)** & **MAE\({}_{\rm px}\) / MAE\({}_{\rm pg}\) / MAE\({}_{\rm pg}\)** \\
**(m)** & **(\(\times\)10\({}^{-16}\)C\(\cdot\)m)** \\ \hline
[550 250 50] & 0.0027 / 0.0030 / 0.0029 & 3.4642 / 3.5243 / 3.6122 \\
[350 150 100 50] & 0.0025 / 0.0025 / 0.0026 & 3.3937 / 3.5066 / 3.5379 \\
[580 280 180 80] & 0.0016 / 0.0017 / 0.0016 & 2.0314 / 2.1723 / 2.0643 \\
[470 370 270 170 70] & 0.0018 / 0.0019 / 0.0019 & 2.3513 / 2.4934 / 2.3510 \\
[350 250 150 100 50] & 0.0018 / 0.0019 / 0.0018 & 2.5087 / 2.5930 / 2.4311 \\ \hline \end{tabular}
\end{table}