## 3 Results  
For evaluation, we split the data into a training and test dataset to fit and test our surrogate models; see Table 6 for the dataset sizes and Table 2 for more details regarding the data split. As a next step, we need to define hyperparameters for each model and each use case. We performed hyperparameter optimization using only training data; no test data were used. In our PINN approaches, the adaptation of hyperparameters was based on the work of [21; 22]. Our MLPs were designed to be similar to our PINNs to allow for fair comparisons. We varied hyperparameters in our neural network approaches (MLP and PINN) following best practices and guidelines, where we optimized the number of hidden layers, number of neurons per hidden layer, activation function, validation split, earlystopping patience and the size of the batch per training epoch. Regarding the rest of our models, we applied a grid-search with a five fold cross-validation, utilizing the training data to obtain the best hyperparameters. The hyperparameters for each use case are in Appendix B and Tables A1-A6. Our evaluation is based on R2-scores with respect to the FEM results and inference time. For models that contain inherent randomness, such as MLPs, GBDTR and PINNs, a five-fold cross-validation was conducted. For these models, we report the mean values and standard deviation of the R2-score. For the sake of brevity, we report only the average R2-scores across all 13 targets in this section; see Tables 7-9. The R2-scores for individual targets are provided in Appendix C. The inference times are based on the mean value of three measurements. Inferences were run on a machine with 16 GB RAM, 8 CPUs and Intel(R) i7-8565 2.0 GHz processor. To compare the inference time of our surrogate models with the computation time required to run FEM simulations, we have included the latter also in Tables 7-9. For graphical results, we chose simulations that cover the error situation quite well in order to make statements about the performance of each model. In addition to the absolute errors (Figures 5a-f-10a-f), the corresponding FEM simulations of the basis are shown in Figures 5g-10g. GBDTR, KNNR, GPR and SVR algorithms were implemented with the scikit-learn library version 0.24.0 in Python. The SVR and GBDTR algorithms were constructed with MultiOutputRegressor scikit-learn API to fit one regressor per target. Regarding our DL algorithms, the utilized MLPs were implemented with the keras API version 2.4.3 and our PINNs were implemented with the sciann API version 0.5.5.0 in Python 3.8.5. We used the PDEs from [21,22], but instead of the inversion part, we trained our PINNs additionally with plastic strain data, same as for the rest of the surrogate models. Figure 5: Elongation of a perforated plate, Simulation 1 (extrapolation): absolute errors of different surrogate models (**a–f**) and ground truth Abaqus FEM simulation (**g**) of \(\sigma_{xy}\). In the elongation of a perforated plate use case, our approach is based on a total of nine FEM simulations. We used five simulations for training and four simulations to evaluate the fitted models; see Table 2. We report the average of R2-scores across all outputs in Table 7 with the corresponding inference times. Regarding extrapolation, the absolute errors of each surrogate model with respect to \(\sigma_{xy}\) of Simulation 1 are shown in Figure 5. We plot the absolute errors of each surrogate model of \(\sigma_{zz}\) of Simulation 4 in Figure 6 as an example of interpolation. In addition, we show in both figures the ground truth of the corresponding output variable obtained from the FEM simulation. For both interpolation and extrapolation, the errors are large near the shear band. As far as extrapolation is concerned, in addition to the errors near the shear band, most models have significant errors near the maximum negative xy shear stresses; see blue areas in Figure 5g.