## 3 Results and Discussion
### Testing the ANN against the Full 3D Simulation Results  
The optimum topology of the ANN is found to have 11 neurons in the first hidden layer and 17 neurons in the second hidden layer. To obtain this topology several ANNs are trained and for each the number of neurons in the hidden layers is being found by systematically testing the ANN performance. The best ANN is chosen such that the model has the highest correlation factor and the lowest relative errors. In this respect, for this best trained ANN the Pearson coefficient and mean relative errors are calculated for the training and validation data subsets and its prediction capabilities are tested considering a subset of data that were not yet seen by the model (i.e., the testing data subset). The excellent correlation between the experimental results (from 3D simulations) and the predicted values is demonstrated by the regression plots and the high Pearson coefficients (Figure 4). Figure 3: **(a)** Sketch of the geometrical arrangement: laser pulse propagates from left to right; interaction cell filled with gas at \(p\) pressure, begins at \(z_{i}\) position relative to focus; \(L_{med}\) is the length of the interaction medium. **(b)** Schematic representation of the parameter space where the full 3D calculations have been performed. For each pulse energy full 3D calculations were performed for several cell positions relative to focus (x axis) and gas pressures (y axis). Red bullets represent the particular cases solved by full 3D simulations and used by ANN for training, validation and testing. The forth scanned parameter is the medium length which is included in every simulation with values 4, 8, 12, 16, 20 cm. From Figure 4 one can observe that the correlation factor has values greater than 0.99 for all data subsets: 0.9998 for the training data subset, 0.9983 for the validation data subset, 0.9979 for the testing data and 0.9996 for the overall data set. As a consequence, the best linear fit of the points and the dotted line that represents the perfect correlation (R = 1) overlap, indicating an extremely reduced deviance of the ANN predictions from the 3D model data. The high performance of the ANN is also demonstrated by the low relative errors which are calculated for the three data sets and are presented in Table 1.  
The maximum value of the relative error is 4.48% and is obtained for the validation data subset. The testing data subset has a similar maximum relative error of 3.94%, whereas the training data subset has a value of 1.8%. In case of the mean relative errors, the testing data subset has the highest relative errors among the three data subsets, with a fairly reduced value of 1.41%. This is expected, as the input testing subset of data was processed by the ANN only after the training-validation step, in order to prove its prediction capacity for the newly met testing data. As a result of the excellent ANN prediction capabilities (extremely low relative errors and high correlation factors), the model was further used to predict the harmonic yield for various input variables. In further applications it is very important to have sufficient flux in a given spectral domain, therefore increasing the harmonic yield is of great interest. We here show that the correctly trained ANN predicts the expected harmonic yield of a possible HHG experiment with the parameters changing in the given range. The ANN works best if the new parameters take values in the range of those used for training and validation. At first glance it may look as a typical interpolation in the multivariable space, however if we consider the complexity of the HHG process given by the highly nonlinear nature of the laser-atom interaction, entangled in the \begin{table}
\begin{tabular}{c c c} \hline \hline  & 1 **Mean Relative Error \(|\) [\%]** & 1 **Maximum Relative Error \(|\) [\%]** \\ \hline Training Data Subset & 0.33 & 1.80 \\ Validation Data Subset & 1.04 & 4.48 \\ Testing Data Subset & 1.41 & 3.94 \\ \hline \hline \end{tabular}
\end{table}