## 2 Collected Data and Research Method
### Modeling of Complex Systems Using Artificial Neural Networks  
Machine learning (ML) comprises a broad range of algorithms, characterized by a high number of tuning parameters that lead to extremely high flexibility in modeling nonlinear systems [24]. Thus, by properly optimizing the algorithm parameters, ML can model and predict the behavior of complex systems where analytical modeling is difficult or computationally unfeasible. Among the different ML approaches, artificial neural networks (ANNs) have emerged as a powerful tool [36; 37]. ANNs emulate the operation of biological neural systems, where different inputs are non-linearly combined. Each neuron on its own performs a simple computation, but when interconnecting a large number of them, the generated ANN acquires a high degree of adaptability, being capable of modeling extremely complex and nonlinear systems. Recent decades have seen an exponential development and sophistication of ANN architectures and applications. The universe of ANNs can then be classified according to many different characteristics: For example, ANNs can be divided into classifiers and regressors, depending on whether the output domain is continuous or discrete. Another classification criterion is the direction of the information flow. Thus, if the information travels always from neurons closer to the input to neurons closer to the output, the ANN is said to be feed-forward [24]. On the contrary, if the network presents any feedback loops, it is denominated as recurrent. Another possible classification relies on the complexity of the ANN architecture, being possible to have either deep or shallow ANNs. The former uses a huge number of interconnected neuron nodes to process the massive amount of data, including text, audio, and image recognition, or natural language processing. Conversely, the latter employs a relatively low number of neurons to adapt, in order to solve simpler problems. For each ANN architecture, the complexity of the network can be tuned, generally, by adding more connections and neuron nodes. In particular, for a given architecture, a low number of neurons results in a simple model that cannot be adapted to the system, whereas an excessively high number of neurons may lead to a complex model that may fit undesired noisy features. This effect, denominated as overfitting, can be avoided by employing regularization terms in the cost function that penalize high numbers of degrees of freedom, or by some dimensionality reduction. Finally, alongside the architecture, the operation mode of the ANN is another important characteristic of its implementation. ANNs can operate in supervised, unsupervised, or reinforced modes. In supervised mode, the ANN is trained (that is, the weights of each neuron are optimized), in order for the output of the network to be the closest possible to the desired data, which are known _a priori_. Therefore, a training set of inputs and their associated outputs must be supplied in supervised mode. This is not the case in unsupervised mode, where the weights of the ANN are optimized by employing some kind of metric computed from the output data, without knowledge of the inputs. The reinforced mode can be understood as a hybrid alternative. Typically, initial, supervised learning is performed to achieve a first approximation of the ANN weights. These weights are then progressively refined using the unsupervised mode. The most suitable architecture and operation mode depends on the problem to be addressed, especially regarding its complexity and available data. For our case, we adopted a swallow feed-forward regressor, operating in supervised mode, to relate the input and output latent variables, as shown in Figure 1. As can be seen, a dedicated ANN is used to