## 1 Introduction
### Feedforward Neural Networks  
Machine learning has become an indispensable tool in many applications. The incorporation of artificial neural networks in complex problems of mechanics has been shown useful in terms of structural response prediction. In this section, we provide the basic idea and key mathematical expressions, summarizing the fundamentals of feedforward neural networks, which can be found in detail in literature [40,41]. The term feedforward neural networks implies already the data flow of the neural network, as the information is transported forward from layer to layer. These architectures comprise at least two layers: an input and an output layer. In this case, they are called single-layer perceptrons. For this type, the output is calculated by the weighted sum of all inputs. However, this architecture is limited to the classification of separable patterns [40]. By adding so-called hidden layers between the input and output layer, one overcomes these limitations. These architectures are called multi-layer or deep feedforward neural networks or multi-layer perceptrons. The simplest architecture has one hidden layer, as depicted in Figure 5. First, one chooses appropriate information parameters for the input layer and the output parameters as the desired targets. In our case, the input parameters are chosen intensity measures, while the output parameter is the failure criterion. The data is passed forward from layer to layer via matrix multiplications, vector additions, and activations, where, in case of supervised learning, the coefficients of these matrices and vectors are the neural network parameters to be found through an optimization process. Figure 4: Acceleration response spectra of all generated earthquakes using the presented procedure. The mean, 25 percentile, and 75 percentile spectrum are compared with the spectrum of the benchmark earthquake (Northridge 1995). Figure 3: (**a**) Accelerogram of the Northridge earthquake [35] in 1994 and (**b**) time history of one sample of the synthetic generated accelerograms using a nonstationary Kanai-Tajimi filter. The procedure of one forward pass is expressed mathematically as follows. The first layer (\(l=1\)), is provided with normalized input data, such as, min-max feature scaling or standard score normalization, cf. [41]. The following layers (\(l>1\)) are fed by the output of the previous layer, \(\hat{y}^{(l)}\). The outputs are weighted by a matrix \(\mathbf{W}^{(l)}\). Furthermore, a bias vector is added \(\mathbf{b}^{(l)}\), i.e., \(\mathbf{z}^{(l)}=\mathbf{W}^{(l)}\mathbf{x}^{(l)}+\mathbf{b}^{(l)}\). Additionally, each neuron within each layer is provided by an activation function \(f(\mathbf{z}^{(l)})\). The most common activation functions are the sigmoid function \(f(z)=\frac{e^{z}}{1+e^{z}}\), the tangent hyperbolicus function \(f(z)=\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}}\), and the rectified linear activation unit \(f(z)=\max{(0,z)}\). The output of every hidden layer is written as \(\hat{\mathbf{y}}^{(l)}=f(\mathbf{z}^{(l)})\). In this manner, the information is transported through all the layers until the output layer \(\hat{y}^{(l=L)}=\hat{y}\). During parameter optimization, the output layer is provided by the targets \(\mathbf{y}\) of the sample. The error is calculated by comparing this target with the output of the last layer. The weights and biases can be optimized by back-propagation of the error [42], e.g., the mean squared error \(\epsilon=\frac{1}{N_{y}}(\mathbf{y}-\hat{\mathbf{y}})^{2}\):  
\[\Delta\mathbf{W}^{(l)}=-\alpha\frac{\partial\epsilon}{\partial\mathbf{W}^{(l) }}\;\;\text{with}\;\;\frac{\partial\epsilon}{\partial\mathbf{W}^{(l)}}=\frac{ \partial\mathbf{z}^{(l)}}{\partial\mathbf{W}^{(l)}}\frac{\partial\hat{ \mathbf{y}}^{(l)}}{\partial\mathbf{z}^{(l)}}\frac{\partial\epsilon}{\partial \hat{\mathbf{y}}^{(l)}}\;. \tag{10}\]  
Hereby, the gradient \(\frac{\partial\epsilon}{\partial\hat{\mathbf{y}}^{(l)}}\) of the last layer (\(l=L\)) leads to \(\frac{2}{N_{y}}(\hat{\mathbf{y}}-\mathbf{y})\).