## 4 Results
### Discussion
#### 4.3.3 Comparison between the PI_DeepONet and Ko Method  
The Ko method also reconstructs the displacement by solving the GDEs of the structure. However, the Ko method solves the GDEs by directly integrating the strain function twice, which necessitates a full-field strain. Therefore, the Ko method is not applicable in situations where no strain measuring points are arranged at the boundary, as we demonstrated in Sections 4.1 and 4.2. In addition, to reconstruct the full-field strain, the Ko method assumes that the strain values between the adjacent strain measurement points are linearly varied, which can lead to a significant reconstruction error in certain scenarios.  
Using the beam with the restrictions of clamped-clamped as the example, the reconstruction performance of the two methods was evaluated. The \(x\) coordinates of the strain measuring points were set as \(\{0.1(i-1)\}_{i=1}^{11}\) (m). After determining the arrangement of the strain measuring points, PI_DeepONet was built and trained for the displacement reconstruction. The strain and displacement under three loading conditions were simulated by FEM, and the displacement was reconstructed by the two methods, using simulated strain as the input. Table 14 summarizes the relative errors of the two methods at the maximum response. For point load \(T_{P}\), both methods have a high reconstruction accuracy. However, for uniform load \(T_{U}\) and staggered load \(T_{S}\), the reconstruction accuracy of the PI_DeepONet is much higher than that of the Ko method. This is because the Ko method assumes that the strain is linearly varied between neighboring strain measuring points. When the structure is subjected to point load, the strain is linearly distributed, so the Ko method achieves high accuracy in this case. However, the assumption of linearly varied strain is not valid when the structure is subjected to complex loads, such as uniform load and staggered load. Thus, the Ko method results in poor reconstruction accuracy in these scenarios. In contrast to the Ko method, PI_DeepONet can extract information about the full-field strain from discrete strains with the assistance of the powerful nonlinear fitting capability of the neural network. Therefore, PI_DeepONet has high reconstruction accuracy, even under complex loading conditions. Furthermore, our proposed method can directly reconstruct the full-field displacement of the beam, which is not achievable via the Ko method.  
\begin{table}
\begin{tabular}{l l l l l} \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Relative Errors**} \\ \cline{2-5}  & \(\mathbf{T_{P}}\) & \(\mathbf{T_{U}}\) & \(\mathbf{T_{S}}\) & \(\mathbf{T_{T}}\) \\ \hline Algorithm 1 & 1.3\% & 0.026\% & 2.0\% & 3.2\% \\ \hline Gaussian probabilistic models & 27\% & 26\% & 64\% & 57\% \\ Normal gradient descent iterations & 9.2\% & 6.7\% & 4.2\% & 21\% \\ \hline \end{tabular}
\end{table}
Table 13: Relative errors at the maximum response calculated by the PI_DeepONet for the plate with the restrictions of clamped–clamped.  
\begin{table}
\begin{tabular}{l l l} \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Relative Errors**} \\ \cline{2-3}  & \(\mathbf{T_{T}}\) & \(\mathbf{T_{S}}\) \\ \hline Algorithm 1 & 1.9\% & 0.35\% \\ \hline Gaussian probabilistic models & 2.8\% & 2.5\% \\ Normal gradient descent iterations & 6.1\% & 3.9\% \\ \hline \end{tabular}
\end{table}
Table 12: Relative errors at the maximum response calculated by the PI_DeepONet for the multi-span beam with the restrictions of clamped–simply supported–clamped.