## 1 Introduction
### The New Training Data Selection Method Based on Sample Intensities  
Artificial neural networks can solve a wide variety of tasks in several fields of application. Once a neural network is properly trained, it can make predictions that are both accurately and efficiently. In general, neural networks will predict reliably as long as they have seen similar patterns before. Thus, one may imagine that the mean of the data set will be excellently predicted. However, in this study, we want to estimate the response of events near the failure region of the distribution, i.e., extreme events that happen very rarely. Therefore, we provide a new strategy to cover the entire domain of possible events during the training procedure. The major steps of the procedure to use machine learning are shown in Figure 6, while the appropriate choice of the training data is addressed in this paper. In particular, Figure 7 shows all the steps of the new selection process. Instead of using an arbitrary training data set \(\mathcal{I}_{t,rand}\), we select a training data set \(\mathcal{I}_{t,sel}\) based on the intensities of the generated artificial earthquake samples. Therefore, we split each intensity range into evenly distributed intervals. Dependent on the number of intensity measures considered for the selection, we choose from an \(i\)-dimensional grid. If the selection is based on one intensity only, we pick one sample from each bin. In this case, the number of selected samples is equal to the number of bins or slightly smaller if empty bins exist. This procedure will likely give a better-distributed training set than a randomly picked set, especially, if the chosen intensity correlates with the output quantity. However, it will probably not cover the whole domain of all intensity measures chosen as input parameters. One may interpret this strategy as an adaption of the Latin-Hypercube sampling approach. The difference is  
Figure 6: Flowchart of machine learning enhanced structural response statistics using the proposed strategy for the selection of the training samples during supervised learning. that we perform the sampling beforehand based on non-stationary Gaussian random white noises and, subsequently, filter the data. The domain is represented better considering two intensities. The procedure can be easily shown for this setup. In Figure 8, we present the correlation between velocity spectrum intensity \(\int_{0.1}^{2.5}S_{v}\) and the spectral displacement \(S_{d}(T_{1})\). In particular, Figure 8a shows all samples from set \(\mathcal{I}\), while Figure 8b shows only 450 samples from this set, in other words a randomly chosen subset used for the training, \(\mathcal{I}_{t,rand}\). As one can see, most samples with high intensities are not included in this set, although they are the most important samples that allow for the neural network to learn more around the tail region of the distribution, as the intensities have a certain correlation with the structural response. In Figure 8c, we also present the samples chosen from \(\mathcal{I}\) using this strategy. Both Intensities were cut into 34 intervals, which resulted in the selection of 438 samples. In doing so, we achieve that the selected training set \(\mathcal{I}_{t,sd}\) covers the whole parameter space of both intensities. Therefore, the neural network will learn from very frequent and from very rare events to the same extent. The crucial part of this strategy is a meaningful choice in the number of the intensities considered and their type to perform the selection procedure. Furthermore, the grid size  
Figure 8: Two-dimensional selection strategy: selection of samples from (**a**) the full set \(\mathcal{I}\) using the velocity spectrum intensity \(\int_{0.1}^{2.5}S_{v}\) and the spectral displacement at the natural period \(S_{d}(T_{1})\), (**b**) randomly chosen samples for the training set \(\mathcal{I}_{t,rand}\), and (**c**) selected samples for the training set \(\mathcal{I}_{t,sd}\).