## 5 Benchmarking of ML Methods and Influence of the Design Space Selected
However, the improvement is reduced for large dataset sizes, and sampling additional points may not justify the increase in accuracy. It is observed that the NN-based surrogates yield better accuracy for all STL models when big datasets are used. This can be justified by the fact that NN can produce a better fit on non-smooth functions than other ML approaches [43]. The GPR model, widely used to construct surrogates as it is well-suited for regression with small databases, showed good performance for the infinite plate and correction factor models; however its performance for highly non-smooth models is hindered as the GPR relies strongly in spatial correlation of the input features [34]. Moreover, the addition of physics-guided features consistently produces more accurate models. The improvement obtained due to the physics-guided features in the NN-based surrogate performance is not as significant as for the other ML models, indicating the high capabilities of NN to learn more complex and relevant representations of the data through its depth even without expert guidance. Except for the FEM model, the NN-based surrogates of all STL models achieved RMSE below 3 dB, which are satisfactory results since measurements errors in STL experiments usually also go up to 3 dB. The RMSEs for the FEM model are higher due to the overall more complex simulation and stochastic excitation, which renders the data less reliable. Another important factor in the choice of a surrogate model is the training time, which is presented in Figure 13. It can be seen that RF and GBT surrogates are significantly faster to train than the other two methods. This is due to the fact that the training of decision-tree-based models scales loglinearly with the dataset size [44]. It is worth noting that because no efficient multi-output implementation of the GBT model is available, it was required to fit one model for each input frequency, thus increasing the overall training time compared to the RF model. Furthermore, GPR is computationally expensive to train for large datasetsas its operations scale as \(\mathcal{O}(N^{3})\)[45]. It is also worth noting that GPR, GBT and RF have sensitivity analysis as a by-product and, therefore, are intrinsically more interpretable than NN. On the other hand, NN and GPR extrapolate better than decision-tree-based methods. Figure 14 shows an example of the predicted STL curve for each STL model using the most accurate surrogate which is the NN model with physics-guided features trained over 2000 supporting points. Visually, the predicted curves are remarkably similar to the simulated ones, indicating that NN-based surrogates can model non-smooth functions from vibroacoustic problems. Figure 15 provides a deeper look into the NN-based surrogate model errors for each STL model, additionally indicating the mean absolute error (MAE) and the mean maximum error (MME). For instance, the MME indicates that high localized errors occur in MS and FEM models without band average. Therefore, although these  
Figure 12: Benchmarking of machine-learning-based surrogate regarding the accuracy in relation to the dataset size: \(\underline{\phantom{\rule{0.0 pt}{1.  
surrogates adequately predict the overall STL behavior, they may considerably mispredicted local maxima and minima of rough response signal. An investigation of the NN-based surrogate performance regarding the set of interest \(\hat{\mathcal{X}}\) was also performed. For that, four design spaces were considered, namely the standard design space \(\mathcal{D}\) from Table 1, the smallest design space \(\mathcal{D}^{-}\) and the biggest design space \(\mathcal{D}_{1}^{+}\) and \(\mathcal{D}_{2}^{+}\), which have the same size but are located in different places in the domain \(\mathcal{X}\), as illustrated in Figure 16. The range of damping, Poisson's ratio, and the width and length of the plates are the same from Table 1 for all design spaces.