## 4 Generation of the Deep Neural Network
### Performance of the DNN  
The data points in Figure 7a,b solely represent the test data set which consists of data points that have not been used for the training the DNN. These samples have been predicted by the neural network and are compared to the exact known values from FEA in two different ways. Figure 5: Visualization of the DNN. Figure 6: Training history of the DNN. Figure 7a shows predicted SCFs for bending and tension versus exact SCFs by FEA. In cases in which a sample was perfectly predicted by the neural network, the data point would lie on the diagonal black line. Figure 7b shows relative errors of the predicted SCFs to the exact SCFs by FEA in the form of two box plots, each boxplot for either tension or bending loading. The relative errors are calculated by:  
\[err_{\text{rel}}=\frac{SCF_{\text{Predicted}}-SCF_{\text{FEM}}}{SCF_{\text{FEM }}}\cdot 100\ [\%] \tag{2}\]  
The middle line within the boxes shows the median value of the relative errors, while the box around this line includes 50% of the data from the 25% to the 75% percentile. The black whiskers outside of the box extend to 1.5 of the interquartile range. The remaining outliers outside of the whiskers are beyond this level. Statistical Characterization of Weld Geometries and Stress Concentrations along Butt Joint Weld Seams  
This method is now applied to determine statistical distributions of weld geometry parameters and SCFs of weld seams of butt joints made by different welding methods. To this end, laser scans of butt joints taken from [54; 55; 56; 57; 58] and other unpublished investigations were collected and assessed using the weld geometry assessment algorithm presented in [PERSON] et al. [15]. In total, 80 welded plates produced by Submerged Arc Welding (SAW), Flux-Cored Arc Welding (FCAW), laser-, and laser-hybrid welding were measured by a laser scanner to obtain the weld geometry along the weld seam. To simplify sampling and to reduce movement-related vibrations of the scanning setup, segments of about 40 mm length were measured in 0.15 mm steps and later combined to reproduce the weld seam. Figure 8 presents the change of weld geometry parameters and SCFs along an exemplary 36 mm weld segment. In the following sections, only the SCF for tension loading is presented, as the results for the bending SCF are similar. Figure 7: Quality of predicted SCFs in the form of: (**a**) scatter plot; (**b**) box plot. On the left side of Figure 8, the change of weld geometry parameters along the weld seam obtained using the curvature method [15; 59] are presented, as well as the SCF under tension loading based on the DNN presented in Section 4. The determined weld geometry parameters are weld toe radii, weld toe angles, and undercut depths on the left and right side of the weld seam, as well as the weld reinforcement height and width. On the right side, histograms of the measured parameters are presented, including four statistical measures to describe the distributions. The four measures are _kurtosis k_, _skewness s_, _median m_, and _variance v_. While the median represents the value for which 50%  
Figure 8: Change of weld geometry parameters and SCFs along exemplary weld seams. of the sample are larger and 50% are smaller, the unbiased or empirical sample variance presents a measure for the spread of data:  
\[v=\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2} \tag{3}\]  
where \(\overline{x}\) is the mean of \(x_{i}\) and \(n\) the number of observations. The skewness is a measure of the asymmetry of the data around the sample mean:  
\[s=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{x_{i}-\overline{x}}{\sigma}\right)^{3} \tag{4}\]  
with \(\sigma\) representing the standard deviation. If the skewness is negative, the data spread to the left, and if the skewness is positive the data spread to the right. The skewness of a symmetrical distribution (e.g., a normal distribution) is zero. Typical distribution functions have a skewness between \(\pm 3\). Values outside this range are generally considered severely skewed.