## 3 Results
### Mapping Using ML Approach
#### 3.5.2 Prediction Results  
From now on, the prediction presented here is solely based on the measurement, as shown in Figure 8a as a test case, and we stop using simulations. We trained the network using only a global and no local contribution (\(\lambda_{1}=1\), \(\lambda_{2}=0\)). We tested the trained FCNN model with our measurement as input, in order to obtain the true shape and position of the coil in our measurement or, in other words, \"detect\" the coil in the measurement. The goal was to achieve a prediction that would resemble, as much as possible, the theoretical magnetic heat map (Figure 5). The trained model prediction is shown in Figure 13.  
Figure 12: Loss function training and validation curves over 200 epochs. \(\lambda_{1}=1\), \(\lambda_{2}=0\).  
We trained the network four additional times with varying global and local contributions: \(\lambda_{1}=0.5\), \(\lambda_{2}=0.5\); \(\lambda_{1}=0.25\), \(\lambda_{2}=0.75\); \(\lambda_{1}=0.1\), \(\lambda_{2}=0.9\); and \(\lambda_{1}=0\), \(\lambda_{2}=1\). Each trained FCNN model was tested using our measurement as input, consistent with the previously stated rationale. The trained model's predictions are shown in Figure 14.  
All the predicted heat maps resemble the theoretical heat map (Figure 5). The magnetic heat map predicted by the FCNN trained with the loss function parameters \(\lambda_{1}=1,\lambda_{2}=0\) (Figure 13) most closely reproduced the shape of the magnetic object. However, the other magnetic heat maps (Figure 14a-d) achieved significantly smaller offsets in the coil's center coordinates, as shown in Table 2 (with the highest offset marked in red and the lowest in blue). Finally, we also calculated the MSE between the theoretical heat map and the predicted heat maps in order to quantitatively estimate the matching of the predicted heat maps to the theoretical heat map, as shown in Table 2.  
Figure 14: (**a**) FCNN prediction with loss function parameters \(\lambda_{1}=0.5\), \(\lambda_{2}=0.5\); (**b**) FCNN prediction with loss function parameters \(\lambda_{1}=0.25\), \(\lambda_{2}=0.75\); (**c**) FCNN prediction with loss function parameters \(\lambda_{1}=0.1\), \(\lambda_{2}=0.9\); (**d**) FCNN prediction with loss function parameters \(\lambda_{1}=0\), \(\lambda_{2}=1\).  
Figure 13: Visualization of FCNN prediction with loss function parameters \(\lambda_{1}=1\), \(\lambda_{2}=0\).  
Intuitively, the prediction that is based on a trained model with loss function that has \(\lambda_{1}=1\), \(\lambda_{2}=0\) should visually be the most similar to the theoretical heat map. As it turns out, the prediction of the trained model with the loss function that has \(\lambda_{1}=1\), \(\lambda_{2}=0\) (Figure 13) is the most similar to the theoretical heat map, by the MSE metric that we used (marked as blue in Table 2), and the prediction of the trained model with the loss function that has \(\lambda_{0.5}=1\), \(\lambda_{2}=0.5\) (Figure 14a) is the least similar (marked as red in Table 2) to the theoretical heat map. We were successful in achieving the proper arrangement of the coil, and the overall topography relative values have also been accurately restored. The final result is quite promising, as the form and general topography now closely resemble the one achieved through a simulation with 1000 sensors (see Figure 9c).