## 5 Benchmarking of ML Methods and Influence of the Design Space Selected  
In this section, surrogate models based on NN, GPR, RF and GBT are evaluated regarding accuracy and training time. The benchmark is conducted for all STL models of Section 2. The influence of physics-guided features is also considered in the test cases. Supporting points are sampled from the design space of Table 1 using LHS. Finally, an analysis of the influence of design space size and location in the surrogate accuracy is performed. The training of the ML models was performed using 80% of the dataset, while the re  
Figure 11: MDI-based sensitivity of the 1/3 octave band average STL of finite plate evaluated by FEM: (**a**) the inputs of the surrogate only consist of the STL variables (\(RMSE=2.23\pm 0.02\) dB); (**b**) physics-guided features \(m\) and \(D_{R}\) are included as input for the surrogate (\(RMSE=2.09\pm 0.02\) dB); (**c**) physics-guided features \(m\), \(D_{R}\) and \(R\) are included as input for the surrogate (\(RMSE=1.63\pm 0.03\) dB). Figure 10: MDI-based sensitivity of the 1/3 octave band average STL of finite plates evaluated by modal summation (MS): (**a**) the inputs of the surrogate only consist of the STL variables (\(RMSE=2.19\pm 0.05\) dB); (**b**) physics-guided features \(m\) and \(D_{R}\) are included as input for the surrogate (\(RMSE=2.01\pm 0.02\) dB); (**c**) physics-guided features \(m\), \(D_{R}\) and \(R\) are included as input for the surrogate (\(RMSE=1.49\pm 0.02\) dB). maining 20% was used as a test dataset to evaluate the surrogate performance using the RMSE obtained with five-fold cross-validation as the evaluation metric. The ML models' hyperparameters were selected based on a trial-and-error search and are as follows:  
_NN-based surrogate models._ The NN were implemented using _Keras_ with _tensorflow_ as backend. One regressor was trained to output all STL predictions at \(n_{outputs}\) frequencies. Standardization was applied to all input features, while the outputs were scaled in the range [0, 1]. The fully connected NNs had an architecture with \(n_{inputs}\) nodes in the input layer, 5 hidden layers with 32 nodes each, and \(n_{outputs}\) nodes in the output layer, where \(n_{inputs}\) is the number of input features. All hidden layers had the _sigmoid_ as activation function and a L2 regularization penalty of \(1\times 10^{-7}\). The training was performed with batches of 32 samples and with 1500 epochs, except for the MS and FEM models, in which the NN trained for 2500 epochs. The Adam algorithm was applied to minimize the average mean squared error of the STL curve prediction. _GPR-based surrogate models._ The kernel \(k\) used to construct the GPR used a constant kernel \(k_{C}\), a Matern 3/2 kernel \(k_{M}\), a radial basis function kernel \(k_{RBF}\) and a white noise kernel \(k_{r}\) combined as \(k(x_{i},x_{j})=k_{C}(x_{i},x_{j})*k_{M}(x_{i},x_{j})+k_{RBF}(x_{i},x_{j})+k_{r }(x_{i},x_{j})\). The kernel hyperparameters were optimized to maximize the log-marginal likelihood of the posterior distribution using the L-BFGS-B algorithm with ten restarts. One Gaussian process regressor predicted the STL at \(n_{outputs}\) frequencies. The preprocessing of the inputs and outputs was the same from the NN-based surrogate models. _RF-based surrogate models._ An RF was trained to output the STL for the entire frequency range analyzed. The RF was made up of 200 CARTs without a depth limit. The impurity criterion was the mean of the averaged squared error among the \(n_{outputs}\) outputs. _GBT-based surrogate models._ The GBT predictor was made up of 125 decision trees with a maximum depth of 10 nodes. The learning rate of the training was set to 0.05. A different GBT predictor was trained for each predicted output willing to minimize the mean squared error. Figure 12 shows the RMSE of each surrogate for all six STL models and different dataset sizes. It can be seen in the results that the surrogates' performance improves considerably when datasets with a size of up to 1000 samples are used.