## 2 Materials and Methods
### Results on the Phase Dataset  
Using the SR to the second WPT allows to derive the formula that describes the system and obtain sufficiently accurate predictions. Although the phase prediction is not perfect, the model can identify the correct trend and obtain acceptable accuracy for an initial estimate. In particular, the simpler model can successfully predict the main behaviour of the function, focusing only on the linear part as shown in Figures 15-17. The time required by this simpler model is just a few minutes. On the other hand, the more complex model achieves more accurate results, correctly identifying also the dependence on \(R\), shown in Figure 18 and approaching to predict the nonlinearity of the problem, shown in Figures 19 and 20. The drawback of this complex model is the time required to train it, which reaches an hour, and even then, these predictions are not yet perfect.  
Figure 14: Real and imaginary parts of impedance value \(Z_{in}\) for different distance values. The image is taken from case 1, with \(\Delta x=0\) mm, \(\Delta y=0\) mm.  
Figure 13: Prediction of the SR model of the distance values. For better visualization, the plot is made for \(S\) parameters at different distance values, but now the dependent variable is the distance. The \(S\) parameter values reported here are in unitary (dimensionless) form. Conversion to decibels (dB) can be achieved using the formula \(S_{dB}=20\log_{10}(|S_{unitary}|)\).  
Figure 16: Phase predicted with symbolic regression using the simple model. Prediction for the load condition of \(R=1000\)\(\Omega\). The prediction is far from the real value, but the main trend is well predicted, giving useful insights for an evaluation of the phase or the best frequency that minimizes the phase shift.  
Figure 17: Phase predicted with symbolic regression using the simple model, for the fixed distance of the coil \(Z_{0}=4\) mm. From this simple prediction, no dependencies with the load condition emerge.  
Figure 15: Phase predicted with symbolic regression using the simple model. Prediction for the load condition of \(R=100\)\(\Omega\). The prediction is far from the real value, but the main trend is well predicted, giving useful insights for an evaluation of the phase or the best frequency that minimizes the phase shift. The time required for the simulation is about five minutes.  
Figure 19: SR prediction of the phase, for fixed load condition \(R=100\)\(\Omega\). The more complex model is used with 500 iterations, 100 populations, binary operators “+”, _”+”, and “/”, and unary operators as the exponential and logarithmic functions. The time required for the simulation is about one hour.  
Figure 20: SR prediction of the phase, for fixed load condition \(R=1000\)\(\Omega\). The more complex model is used with 500 iterations, 100 populations, binary operators “+”, _”+”, and “/”, and unary operators as exponential and logarithmic functions. The time required for the simulation is about one hour.  
The formula extracted with the simpler model is  
\[\phi(f,R,z_{0})=-\alpha f+\phi_{0}+\beta z_{0}\, \tag{3}\]  
with \(\alpha=0.0015\), \(\phi_{0}=137.40\) and \(\beta=3.49\).  
As shown also in Figures 15-17, this formula is quite simple but still provides useful insights into the original data, catching the correct behavior of the function with good-enough accuracy in the prediction.  
With the more complex model, the formula extracted is more complex, with the same linear component as Equation (3), and a term having the logarithm of \(R\) times \(z_{0}\). Reporting it here, is not considered to be of interest.  
These results show good enough results for a first prediction to improve the control system proposed in [44], and show how SR could be helpful in understanding the behavior of this WPT system.