## 1 Introduction and Motivation
### Artificial Neural Networks  
In an ANN, neurons are the ones responsible for processing the information. Each computing unit receives a weighted signal from all the previous neighboring neurons, sums the received signals and its own bias, processes the information via a transfer function and generates the output [27; 56]. The neurons are organized in input layer, output layer and one or several hidden layers. As general feature, for prediction purposes, the connections between the neurons that belong to the same layer are not allowed, the information being directed from the input towards the output layer [57; 58]. This kind of ANN is named feed-forward neural network and, in most cases, it is associated with the back-propagation learning algorithm [26; 59; 60]. The main steps performed for training the ANN using the backpropagation algorithm are presented in Figure 1a. Input and desired output data sets are firstly scaled into the [\(-\)1, 1] interval for computation accuracy and then divided in training, validation and testing subsets. The backpropagation algorithm, presented in the detail of Figure 1b, iteratively adjusts the weights and biases of the ANN, starting from random values, until the difference between the ANN computed and desired outputs (error) is less than an imposed threshold. In order to avoid overfitting, the validation subset is used to early stop the training procedure. The trained ANN performance is then assessed using the testing data subset, not yet seen by the ANN. Regarding the HHG process, it is known that the harmonic yield is non-linearly influenced by several parameters such as the fundamental laser pulse characteristics, focusing geometry, gas type and pressure which characterize the medium where HHG takes place. For this particular study we chose as varying parameters the laser pulse energy (\(E\)), gas pressure (\(p\)), gas cell position (\(z_{i}\)) and medium length (\(L_{med}\)). Consequently, these four process parameters are chosen to be the ANN inputs, whereas the harmonic yield is the model output. Based on the 3D non-adiabatic model, a total number of 453 input-output data sets were obtained and, subsequently, were divided in: (i) training data which represent 77.9% of the total amount of input-output data sets, (ii) validation data (15.7%) and (iii) testing data (6.4%). The testing data subset is obtained by uniformly extracting every sixteenth sample from the initial data. From the remaining data, every sixth sample was extracted to obtain the validation subset, whereas the remaining data represent the training data subset. The higher percentage of the training and validation subsets with respect to the testing subset is explained by the fact that the process is extremely nonlinear and, for good training results, more data are needed for the training and validation steps. As a result of the complexity of the HHG process, the multilayer feed-forward network is structured in one input layer, two hidden layers and one output layer, the appropriate number of neurons in the hidden layers being found by systematically testing the ANN performance (Figure 2). The information from the hidden and output layers is processed with two different transfer functions: the hidden layer uses a tansigmoid function (Equation (2)) [61], whereas a linear transfer function is applied for the output layer. The ANN is trained with the Levenberg-Marquardt backpropagation algorithm, which is a standard technique for solving nonlinear least square problems [62; 63], is included in the backpropagation procedure. \[f=\frac{2}{1+\exp(-2n)}-1 \tag{2}\]  
where \(n\) is the sums of the neuron received signals and its own bias. Figure 1: (**a**) Flowchart of the computation, (**b**) Flowchart of the backpropagation algorithm. Figure 2: ANN architecture and the schematic representation of a neuron. Several networks are trained and the best is chosen such that the ANN performance is maximized.