## 3 Results and Discussion
### Prediction Potential of the ANN
Even if we use logarithmic scale, the yields predicted with 9 mJ are very close to those obtained with 11.6 mJ. The explanation for this situation can be found in the self-guiding propagation of strong laser pulses in ionizing medium. This result can be good news for attosecond science infrastructures where the other parameters--including cell position and length--are tunable, but the available laser power is limited. Particularly, the ANN simulation was able to reproduce the transition between two different regimes of laser pulse propagation and consequently different conditions for HHG. At low pulse energy (i.e., starting intensity) and also low pressure conditions we find two comparable maxima of conversion efficiency--see data in Figure 5a with the cell positions before and after the focus. The harmonic yield is calculated with Equation (1), where the radial integration guaranties that off-axis contributions are also taken into account. In these mild macroscopic conditions the fundamental pulse does not suffer serious distortions during propagation, therefore we find a confirmation of the phase-matching model of [PERSON] [65; 66]: in the converging beam harmonics find favorable phase-matching conditions off-axis; while in the diverging beam collinear phase-matching on-axis helps the constructive build-up of the generated harmonic field. The situation is different at higher pressure as seen in Figure 5b. The maximum conversion efficiency found in the whole explored multi-dimensional parameter space has been found at \(E\) = 11.6 mJ, \(p\) = 5.31 mbar, \(z_{i}\) = \(-\)25 cm, \(L_{med}\) = 16 cm. The maximum harmonic yield in Figure 5b (\(p\) = 5.31 mbar) is two orders of magnitude higher than the maximum yield in Figure 5a (\(p\) = 0.38 mbar). The physical reason for this behavior is that the conditions for the formation of a self-guided beam propagation are fulfilled. A stable \"working intensity\" is maintained in a large interaction volume (both axially and radially) which creates good phase-matching conditions for a range of harmonics. The conditions that have to be fulfilled in order to have plasma-core induced self-guiding, as well as its effect on the phase-matching of HHG was also studied [67], but goes beyond the scope of this paper. The ANN has no knowledge about the physics of HHG, however it has been trained with data that contain synthetically the whole nonlinear process. It is highly recommended to perform the complete 3D simulation in cases when ANN predicts less intuitive output. In Figure 6 we present the main results obtained with the full 3D model for the parameter combination (9 mJ; 5.31 mbar; \(-\)25 cm; 20 cm). The power spectrum in Figure 6a confirms that the highest yield is expected towards the end of the interaction region. In Figure 6b we show the spatial (\(r\),\(z\)) evolution of the laser pulse's peak intensity, which confirms the formation of a self-guided propagation regime with stable peak intensity 2.5-3 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\). Figure 6c shows the spatial (\(r\),\(z\)) evolution of the harmonic order H25. In accordance with the power spectrum, it is confirmed that H25 builds up constructively and attains maximum yield towards the end of the interaction region, predominantly off-axis. In correlation with the evolution of the fundamental pulse (Figure 6b) the spatial region of maximum H25 intensity is the same where the off-axis refocalization of the fundamental beam happens. In the absorption limited HHG regime the harmonic yield (i.e., the output photon number) should grow quadratically with the optical density under given phase-matching conditions [64]. In Figure 7 we represent the harmonic yield as function of the optical density of the medium. Symbols are data coming from the full 3D simulations, lines are the results predicted by the ANN. With dashed line we show as guide to the eye the line with slope = 2.