## 3 Results and Discussion
### Prediction Potential of the ANN  
The quantity we propose to predict as the main outcome of a virtual HHG experiment is the logarithm of the harmonic yield (Equation (1)) in the specified spectral domain, namely in the 20-40 eV region (between harmonic orders H13-H27). We have chosen this quantity because it can be directly related to the total photon flux in the specified spectral domain which is of central interest in any attosecond science experiment. Since the harmonic yield spans several orders of magnitude, it was more convenient in the training procedure to take its logarithm as the output quantity. One very important parameter in a HHG experiment is the optical density of the generation medium. In the usual cases the length of the interaction medium is constant, thus the optical density is proportional to the gas pressure. However, since we have as scanning parameter the medium length as well, we here use \(p\times L_{med}\) as parameter, a quantity which is proportional to the optical density. In the absorption limited HHG regime the harmonic yield (i.e., the total photon number) should grow with the square of the optical density in a configuration with fixed phase-matching conditions [64]. In order to have better insight into the physics of how the harmonic yield evolves with the optical density of the medium, we show the results as a function of the \(\log_{10}(p[\mathrm{mbar}]\times L_{med}[\mathrm{cm}])\). In Figure 5 we show the prediction capability of the ANN. Dots represent full 3D calculations, while the \"+\" symbols are the ANN predicted yields for two intermediate pulse energies, namely 6 mJ and 9 mJ. In Figure 5 we represent the expected evolution of the harmonic yield as a function of the gas cell position relative to focus for different laser pulse energy values, while the gas pressure and medium length are fixed: (a) \(p\) = 0.38 mbar and \(L_{med}\) = 20 cm; (b) \(p\) = 5.31 mbar and \(L_{med}\) = 20 cm. Comparing the data in (a) and (b), we expect that the ratio of the yields for the same pulse energy should be close to the ratio of the square of the pressures (around 200) which is clearly not the case. This is because the cell position is a critical factor in every HHG process, influencing the yield via propagation of the fields, phase-matching effects and absorption of the harmonics. The prediction potential of the ANN simulation proves to be excellent, and captures the highly nonlinear dependence of the macroscopic harmonic yield on many parameters. Especially important conclusion can be drawn from the position-dependence of the yield, namely that very high input pulse energy is not in all cases necessary in order to obtain high harmonic efficiency. Harmonic yield predicted with \(E\) = 9 mJ pulse is almost as high as those obtained from 3D simulations with \(E\) = 11.6 mJ  
Figure 5: Harmonic yield as function of the gas cell position relative to the nominal focus. Dots are data obtained from the complete 3D simulations, “+” symbols are the results of the ANN predictions. Panel (**a**) synthesizes the results obtained for different laser pulse energies at the fixed parameter values \(p\) = 0.38 mbar and \(L_{med}\) = 20 cm, i.e., at the cell end. Panel (**b**) shows the results obtained with \(p\) = 5.31 mbar and \(L_{med}\) = 20 cm. Colors of the symbols dots and “+” are the same for the same energy. At the intermediate energies, 6 and 9 mJ, we have performed merely a few but computationally demanding 3D simulations and only with \(p\) = 5.31 mbar. pulse energy. In order to correctly interpret the results we compare the full 3D simulation results with the ANN predictions as presented in Figure 5b. Magenta dots are yields from 3D simulations and agree well with magenta \"+\" symbols which are yields predicted by the ANN, being a signature of the ANN high performance. Blue dots are yields obtained from 3D simulations with 11.6 mJ pulse energy. At the zi = \(-\)25 cm starting position the ANN predicts the same high yield also for 9 mJ input energy.