## 4 Sensitivity Analysis of Random Forest Surrogate and Influence of Physics-Guided Features  
In this section, the consistency of the ML-based surrogates with physics-driven models of STL is investigated. MDI-based sensitivity analysis is used to demonstrate the importance of each parameter in the system response, rendering the surrogate more interpretable. The addition of two physics-guided features to the RF models, namely the mass density \(m\) and the real part of the bending stiffness \(D_{R}\), is also evaluated. According to STL theory, the mass density controls the STL in the mass region, while bending stiffness has a significant influence on both the stiffness and coincidence regions. The database used has 2000 supporting points sampled with LHS from the design space of Table 1, which encompasses a wide range of aluminum grades and aluminum matrix composites. The critical frequencies for this design space range between approximately 1000 and 2500 Hz, as shown in Figure 4.  
One RF regressor was trained for each frequency of interest to obtain the features' importance for each output frequency. Each RF was trained with 200 decision trees without a depth limit and with MSE as the impurity criterion. A desktop with a hexa-core 3.1 GHz processor and 32 GB of RAM was used to run the training of the surrogates. The root mean squared error (RMSE) was used to compare the accuracy of surrogates with and without physics-guided features as input. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(\rho\) & \(E\) & \(\
u\) & \(\eta\) & \(h\) & \(a\) & \(b\) \\  & kg/m* & **GPa** & **-** & **\%** & **mm** & **m** & **m** \\ \hline
**min** & 2000 & 60 & 0.25 & 0.1 & 5 & 0.3 & 0.3 \\
**max** & 3000 & 150 & 0.35 & 2.0 & 7 & 0.6 & 0.6 \\ \end{tabular}
\end{table} 
Table 1: Design space with set of interest used to sample supporting points. Figure 5a,b show the MDI-based sensitivity index obtained with the infinite plate models with and without the use of physics-guided features, respectively. The mass-controlled region is distinctly observable at low frequencies in both cases. Approaching the coincidence region, the STL is controlled mainly by the plate density, and the thickness importance is negligible. Despite the damping factor determining the STL amplitude at the critical frequency, Figure 5b shows that the real part of the bending stiffness is the most significant feature at this frequency range. As noted in [26], when a set of interest rather than a specific plate design is considered, the location of the critical frequency dip is more relevant for the STL result than its amplitude, which explains this result. At higher frequencies in the coincidence region, the damping importance increases because for each frequency \(\omega>\omega_{crit}\) there will be a component of the diffuse field with an angle \(\theta\) such that \(\omega_{coinc}(\theta)=\omega\), and the amplitude is again the major aspect of the STL. Finally, it is noted that the inclusion of the physics-guided features improves the surrogate accuracy from an RMSE of 0.21 \(\pm\) 0.00 dB to 0.12 \(\pm\) 0.01 dB. The sensitivity indices of STL of finite plates based on the correction factor approach (Figure 6) differ from the infinite plate indices only in the low-frequency region. This result is coherent with the methodology, which applies a correction factor in the low-frequency range to account for the plate dimensions and comprehend its resonant behavior. Although the plate stiffness would theoretically also impact the resonant modes of the structure, the STL evaluated by the correction factor is insensitive to this feature in the resonant region of Figure 6. Once more, the RMSE decreases from 0.36 dB to 0.25 dB with the inclusion of physics-guided features.