## 3 Simulation Results and Discussions
### Verification Results

Table 2: Test results for various hidden layers and neurons in the case of 40,000 data samples. Figure 5: RE (of the test sample) between the generated and the predicted values of the electric field components E\({}_{x}\), E\({}_{y}\), E\({}_{z}\), across the 14 measurement points and LR = 0.01. Figure 6 illustrates the deviation between the test generated and calculated (output of the neural network) values of the position and electric moment, of the test samples (10%) for the case of 40,000 data samples and the learning rate \(LR=0.01\). Figure 7 presents the corresponding differences in the electric moment values. It should be noted that in the majority of the test samples, the difference between the predicted and the initially generated samples' position is smaller than 5 mm, while more than 40% of the test samples had a maximum difference in position of 2 mm per axis. Moreover, the difference in the electric moments of the generated dipoles and the predicted ones was also smaller than \(5\times 10^{-16}\)C\(\cdot\)m, while 40% of the test sample had a difference of less than \(2\times 10^{-16}\)C\(\cdot\)m, rendering a notable accuracy. Moreover, a complexity calculation was performed indicatively for (i) the DE algorithm of [2], and (ii) the proposed ANN, according to [16]. For a fully connected feedforward network (also known as a multi-layer perceptron), the time complexity of the inference process is O(N), where N is the total number of weights in the network. This is because each weight participates in exactly one multiplication and one addition operation during the forward propagation. This way, assuming that all layers have the maximum number of 580 neurons, the worst-case complexity of our network is O(L \(\times\) n \(\times\) m) = 97,440, where L is the number of layers, n is the number of neurons, and m is the input dimension. On the other hand, for the DE, the time complexity is O(D \(\times\) NP \(\times\) GMAX)= 2,688,000, where D is the input dimension (3 \(\times\) 14 = 42), NP is the population number (80), and GMAX is the number of iterations (800). Based on these worst-case calculations, the proposed ANN achieves more than 25 times faster source prediction than the DE algorithm. Tables 4 and 5 present the generated and the calculated (by the neural network) 10 first values of x, y, and z electric dipole position values of the test sample for the case of 40,000 data samples and for the optimal neural network structure with four layers and 580, 280, 180, and 80 neurons at each layer. The aforementioned errors in Table 1 can be easily observed in the presented examples. Figure 6: Difference between generated and calculated values of (**a**) x dipole position, (**b**) y dipole position, and (**c**) z dipole position. Figure 7: Difference between generated and calculated values of (**a**) p\({}_{\rm x}\) dipole electric moment, (**b**) by dipole electric moment, and (**c**) p\({}_{\rm z}\) dipole electric moment and for LR = 0.01. Next, Figure 8 depicts the neural network performance (Mean Square Error) for the training, test, and validation data during the training. As can be observed, training was completed in 76,102 epochs and the three curves (training, validation, and test) converge almost in the same value of the Mean Square Error.