## 3 Numerical Example
### Predicted Response Statistics  
The response statistics using the Crude Monte Carlo method were performed calculating \(I=10^{4}\) artificially generated earthquakes, which constitutes for all simulations the entire benchmark set \(\mathcal{S}\). The peak story drift ratio set \(\mathcal{O}\) was calculated using the finite element method by evaluating the reference solutions through numeric time integration using the Newmark method [54]. We applied the new strategy to the structures A, B, and C (see Figure 11) using the proposed sample selection for two, three, and six intensities, respectively. For each selection, an artificial neural network was trained. The input vector \(\mathbf{X}\) consists of six features; therefore, the input layer also had six neurons. We used three hidden layers, consisting of between 11 and 14 neurons, using rectified linear activation functions. The output layer consisted of one neuron only, which produced the PSDR prediction. We used the ADAM algorithm to update the learning rate during training [55]. This configuration was used for all neural network architectures presented in this study. The statistics of the predictions are shown in Figure 14. For the structures, A, B, and C, the prediction of the PSDR is shown in the top, middle, and bottom plots of Figure 14, respectively. The probability density function (PDF) is shown on the left-hand side, while the complementary cumulative distribution function (CCDF) is shown on the right-hand side. For the two-dimensional selection process, the samples were selected based on the velocity spectrum intensity \(\int_{0.1}^{2.5}S_{v}\) and the spectral displacement \(S_{d}(T_{1,A})\) using 29 bins each. The selected set consists of \(n=333\) samples, which was used for the training. The validation data was taken from the full set and has a size of 20% of the entire training set. Furthermore, considering the spectral acceleration \(S_{d}(T_{1,A})\), the 3D selection method used 15 bins, which resulted in 336 samples for the training. The six-dimensional selection method used five bins, which resulted in 291 samples. The PDF of the predicted PSDR of structure A is shown in Figure 14a and shows the expected agreement with the finite element solution. In Figure 14b, the corresponding CCDF is shown. On the one hand, we can see that the prediction of randomly selected data cannot predict the tail region correctly. On the other hand, all neural networks trained by the new selected data approach predict this region well. We performed the same procedure for structure B. The two-dimensional selection process was based on \(\int_{0.1}^{2.5}S_{v}\) and \(S_{d}(T_{1,B})\), using 27 bins for each resulted in \(n=318\) training data samples. Considering the spectral acceleration \(S_{a}(T_{1,B})\) for the 3D selection, we used 15 bins, which resulted in \(n=325\) samples for the training set \(\mathcal{S}_{t,sel}\). We used only five bins for the six-dimensional selection, which resulted in \(n=274\) samples. Independent of the training data, the PDF of the predicted PSDR of structure B, shown in Figure 14c, matches the finite element solution. Figure 14d shows the corresponding CCDF. Compared to the previous example using structure A, we observe that the prediction of randomly selected data performs better in the tail region. This structure shows higher resistance to earthquakes and the neural network can predict the results of the extreme event better in this case, although the prediction overshoots the finite element solution slightly. The neural networks trained by the selected data approach can predict this region well. Structure C shows the highest share of samples exceeding a PSDR of 4.5%. The two dimensional selection process based on \(\int_{0.1}^{2.5}S_{v}\) and \(S_{d}(T_{1,C})\), with 28 bins each, resulted in \(n=331\) selected samples for training. Considering \(S_{a}(T_{1,C})\) for the three dimensional selection and using 14 bins, resulted in \(n=309\) selected samples.