## 4 Sensitivity Analysis of Random Forest Surrogate and Influence of Physics-Guided Features
Figure 4: Histogram of the critical frequencies from the STL designs in the set of interest from Table 1.  
Figure 5: The mean decrease in impurity (MDI)-based sensitivity indices of the STL with the analytical infinite plate model: (**a**) the inputs of the surrogate only consist of the STL variables (\(RMSE=0.21\pm 0.00\) dB); (**b**) physics-guided features \(m\) and \(D_{R}\) are included as input for the surrogate (\(RMSE=0.12\pm 0.01\) dB). The MDI-based sensitivity indices change considerably when the model thoroughly considers the structural dynamics of the finite plate and its resonant modes, as for the STL results with MS and FEM approaches in Figures 7 and 8, respectively. Overall, the sensitivity indices of the MS and FEM approaches are similar to each other, indicating that they model the vibroacoustic phenomena with analogous considerations. The bending stiffness and the plate dimensions have a considerable high sensitivity index throughout the frequency range. Therefore, the behavior of the finite plates under analysis diverges from the one of a limp mass, an assumption held by the mass law and considered in the mass-controlled region of both the analytical approach for infinite plates and the correction factor approach for finite plates. Indeed, the mass density importance is negligible in the resonant region, and its influence in the middle frequency range is not predominant. The achieved surrogate accuracy with physics-guided features is \(3.24\pm 0.02\) dB for the MS-based results and \(5.60\pm 0.06\) dB for the FEM-based results. The errors are higher for the FEM model than for the MS model since the excitation in FEM is modeled by random components, and therefore, the results are non-deterministic. The high complexity and non-smooth behavior of the STL response with ML and FEM approaches lead to poor surrogate accuracy even when the physics-guided features are used, which also compromises the accuracy of the sensitivity analyses. In particular, at high frequencies, the even distribution of the sensitivity indices among the features indicates higher complexity and possibly higher inaccuracies. As the modal behavior of the plate is posing difficulties for the surrogate model, the resonance coefficient term \(R\) in Equation (36) is included as another physics-guided feature for the surrogates of the FEM and MS models. The resonant term  
Figure 6: MDI-based sensitivity of the STL of finite plates evaluated with the correction factor: (**a**) the inputs of the surrogate only consist of the STL variables (\(RMSE=0.36\pm 0.01\) dB); (**b**) physics-guided features \(m\) and \(D_{R}\) are included as input for the surrogate (\(RMSE=0.25\pm 0.01\) dB). Figure 7: MDI-based sensitivity of the STL of finite plates evaluated with modal summation: (**a**) the inputs of the surrogate only consist of the STL variables (\(RMSE=3.38\pm 0.03\) dB); (**b**) physics-guided features \(m\) and \(D_{R}\) are included as input for the surrogate (\(RMSE=3.24\pm 0.02\) dB). is the coefficient of the first term of Equation (30) and relates with the simply supported plate natural frequencies. \[R=\frac{D}{m(a^{4}b^{4})}. \tag{36}\]  
As observed in Figure 9, the resonance coefficient feature controls the STL in the low-frequency range. Although the accuracy improved to \(2.67\pm 0.02\) dB for the MS-based results and to \(4.86\pm 0.05\) dB for the FEM-based results, it remains elevated. Extra improvements in the accuracy could be obtained by feature selection, that is, removing unimportant features, a common technique in ML. Finally, the MDI-based sensitivity index is evaluated with one-third octave band average STL with MS and FEM, as shown in Figures 10 and 11, respectively. Once again, the stiffness and dimensions of the plate play a major role for all frequencies. The surrogate models that include the three physics-guided features are the ones with the best accuracy, which is \(RMSE=1.49\pm 0.02\) dB for the MS model and \(RMSE=1.63\pm 0.03\) dB for the FEM model.