## 3 Results
GBDTR performs well overall, though the error increases in  
Figure 6: Elongation of a perforated plate, Simulation 4 (interpolation): absolute errors of different surrogate models (**a–f**) and ground truth Abaqus FEM simulation (**g**) of \(\sigma_{zz}\). various locations; while PINNs have a similar average performance, they perform better outside the shear band regarding absolute errors. MLP overall shows the best results followed by KNNR. In the bending beam use case, similar to the perforated plate use case, we trained our models on five simulations and tested them using the remaining four, see Table 2. We present the average R2-scores across all outputs and inference times in Table 8 for the test simulations 1, 4, 6 and 9. We provide a graphical representation of the absolute error of the surrogate models regarding \(\varepsilon_{yy}^{t}\) in Figure 7a-f with the FEM simulation result in (g) as one instance of interpolation. Absolute errors of the surrogate models regarding \(\varepsilon_{xx}^{p}\) and extrapolation are shown in Figure 8. Overall higher errors can be observed near the encastred boundary condition of the beam for some models for that output. While the PINN shows a competitive average R2-score regarding interpolation, on this single target, its performance shows significant weaknesses. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Model** & **MLP** & **PINN** & **SVR** & **GBDTR** & **KNNR** & **GPR** & **FEM** \\ \hline \multirow{4}{*}{R2} & \multicolumn{6}{c}{Simulation 1} \\ \cline{2-7}  & 0.6682 & 0.6165 & \multirow{2}{*}{0.5122} & **0.7120** & **0.7120** & \multirow{2}{*}{0.6288} & \multirow{2}{*}{0.5377} & \multirow{2}{*}{-} \\  & (7.345 \(\times\)\(10^{-6}\) ) & & & & & \\ \cline{1-1}  & 0.0781 & 0.0638 & 1.20 & 0.271 & 0.00734 & 0.139 & 9.08 \\ \cline{1-1} \cline{2-7}  & \multicolumn{6}{c}{Simulation 4} \\ \cline{1-1} \cline{2-7}  & **0.9979** & 0.9379 & \multirow{2}{*}{0.7558} & 0.9640 & \multirow{2}{*}{0.9621} & \multirow{2}{*}{0.8243} & \multirow{2}{*}{-} \\  & (1.319 \(\times\)\(10^{-3}\) ) & & & & & \\ \cline{1-1} Inference time [s] & 0.0457 & 0.110 & 0.418 & 0.0541 & 0.0790 & 0.221 & 6.81 \\ \cline{1-1} \cline{2-7}  & \multicolumn{6}{c}{Simulation 6} \\ \cline{1-1} \cline{2-7}  & **0.9981** & 0.9314 & \multirow{2}{*}{0.7406} & 0.9516 & \multirow{2}{*}{0.9617} & \multirow{2}{*}{0.8059} & \multirow{2}{*}{-} \\  & (8.315 \(\times\)\(10^{-4}\) ) & & & & & \\ \cline{1-1} Inference time [s] & 0.0442 & 0.0668 & 0.402 & 0.0569 & 0.0705 & 0.212 & 6.61 \\ \cline{1-1} \cline{2-7}  & \multicolumn{6}{c}{Simulation 9} \\ \cline{1-Figure 7: Bending of a beam, Simulation 6 (interpolation): absolute errors of different surrogate models (**a–f**) and ground truth Abaqus FEM simulation (**g**) of \(\epsilon_{yy}^{t}\). The compression of a block with four perforations use case presents a more complex setting because we generalize by two generalization variables (yield stress and block width). Therefore, we utilize more training data for this use case; see Table 2. We report the average results of R2-scores with corresponding standard deviations, if applicable, in Table 9.  
Figure 8: Bending of a beam, Simulation 9 (extrapolation): absolute errors of different surrogate models (**a–f**) and ground truth Abaqus FEM simulation (**g**) of \(\varepsilon_{xx}^{p}\). As an instance for interpolation, the absolute errors regarding \(\varepsilon_{xx}^{p}\) can be seen in Figure 9a-f with Abaqus FEM simulation result (g). Respectively, an instance for extrapolation is shown in Figure 10 with absolute errors (a-f) and FEM ground truth (g). Some models show higher prediction errors near shear bands (high \(\varepsilon_{xx}^{p}\) regions) regarding the interpolation task. However, SVR and GPR cannot extract meaningful information from the training data, especially in the space free of plastic deformation. This is indicated by the low average R2-scores, compared to the other models.