## 4 Analysis of the Inversion Results
### The Inversion Calculation of the Dynamic Modulus and Depth for Pavement Structures  
As with the dynamic modulus, the thickness of the pavement structures also has a significant effect on the pavement deflection. In this example, the same pavement profile as in Section 4.3 was used with the assumption that, in addition to the dynamic modulus, the thicknesses of the three pavement structure layers were also unknown. Hence, the number of the unknown inversion variables is twice as large as the example in Section 4.3. In order to obtain an accurate result, more samples are needed for the training than the previous example. A total of 3000 samples are calculated, of which 2400 samples were used for training, 300 samples for validation, and 300 samples for testing. The dynamic modulus and thickness are randomly generated in these 3000 samples. The value ranges of these inversion variables are shown in Table 8.  
Figure 9: The RMSE and loss change curve of the training samples.  
The inversion results of the validation samples were compared with the exact values, as exhibited in Figures 12 and 13. Figure 12a-c represent the comparison results of the dynamic elastic moduli \(E_{1}\), \(E_{2}\), and \(E_{3}\) of the pavement structure layers, respectively. Figure 13a-c represent the comparison results of the dynamic elastic moduli \(h_{1}\), \(h_{2}\), and \(h_{3}\) of the pavement structure layers, respectively. The average relative error of the six inversion variables is listed in Table 8. As the figures show, the predicted modulus and depth obtained by the presented inversion algorithm have a good agreement with exact modulus and depth. The non-unique problem in the inversion calculation of the pavement structure parameters is solved by the proposed algorithm in engineering accuracy requirements.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Inversion Variables** & \(\mathbf{E_{1}}\) & \(\mathbf{E_{2}}\) & \(\mathbf{E_{3}}\) & \(\mathbf{h_{1}}\)**(m)** & \(\mathbf{h_{2}}\)**(m)** & \(\mathbf{h_{3}}\)**(m)** \\ \hline Value range & 1500–4500 & 150–450 & 75–225 & 0.075–0.225 & 0.15–0.45 & 0.30–0.90 \\ average relative error & 7.96\% & 3.52\% & 6.90\% & 10.58\% & 8.51\% & 11.97\% \\ \hline \hline \end{tabular}
\end{table}
Table 8: The value range of the inversion variables.  
Figure 10: The curves of the predicted and measured values of the testing samples. (**a**) Dynamic elastic modulus of the first layer of the pavement structure; (**b**) Dynamic elastic modulus of the second layer of the pavement structure; (**c**) Dynamic elastic modulus of the third layer of the pavement structure.  
Figure 11: The relative error curve of the testing samples. (**a**) Relative error of the first layer of the pavement structure; (**b**) Relative error of the second layer of the pavement structure; (**c**) Relative error of the second layer of the pavement structure.  
Figure 12: Comparison between the predicted and exact dynamic modulus of the testing samples. (**a**) Dynamic elastic modulus of the first layer of the pavement structure; (**b**) Dynamic elastic modulus of the second layer of the pavement structure; (**c**) Dynamic elastic modulus of the third layer of the pavement structure.