## 4 Results
### Plate
#### 4.2.1 Displacement Reconstruction for Rectangular Plate  
In this section, a rectangular thin plate is utilized to verify the reconstruction performance of PI_DeepONet. The plate is restricted at two opposite sides and free on the remaining two sides. The restrictions applied to the plate are one side clamped, simply supported-simply supported, clamped-simply supported, and clamped-clamped, respectively. Figure 16 shows the geometric model of the plate with one side clamped. There are 30 strain measuring points arranged on the three parallel lines along the \(x\) coordinate direction. The \(y\) coordinates of the three parallel lines are (0.02 m, 0.25 m, 0.48 m), respectively. Figure 14: Reconstruction results of the displacement and rotating angle of the multi-span beam with the restrictions of clamped–simply supported–simply supported. Figure 15: Reconstruction results of the displacement and rotating angle of the multi-span beam with the restrictions of three points simply supported. \begin{table}
\begin{tabular}{l l l l l} \hline \hline \multirow{2}{*}{**Restrictions**} & \multicolumn{2}{c}{**Displacement Fitting Accuracy**} & \multicolumn{2}{c}{**Rotating Angle Fitting Accuracy**} \\ \cline{2-5}  & \(\mathbf{T_{T}}\) & \(\mathbf{T_{S}}\) & \(\mathbf{T_{T}}\) & \(\mathbf{T_{S}}\) \\ \hline clamped–simply supported–clamped & 0.9983 & 0.9999 & 0.9986 & 0.9994 \\ clamped–simply supported–simply supported & 0.9998 & 0.9998 & 0.9998 & 0.9997 \\ three points simply supported & 0.9999 & 0.9999 & 0.9998 & 0.9998 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fitting accuracy of the PI_DeepONet for the multi-span beam. The \(x\) coordinates of the strain measuring points on each line are \(\left\{0.02+0.01(i-1)\right\}_{i}^{10}\) (m). All the arranged measuring points modeling FBG-based strain gauges detect the strain in the \(x\) coordinate direction. The strain-displacement relation and restrictions along three parallel lines share the GDEs of the unified form. The GDEs along the parallel lines were used for formulating the physics losses for PI_DeepONet. For plates with different restrictions, a corresponding PI_DeepONet was built and trained separately to reconstruct the displacement. The strain and displacement were simulated by FEM under four loading conditions. The loads applied to the plate are shown in Table 6. Inputting the simulated strain and the coordinate of displacement measuring point on each line, the reconstructed displacement at the displacement measuring point can be outputted by the PI_DeepONet. The \(x\) coordinates of the displacement measuring points on each line are \(\left\{0.05(i-1)\right\}_{i=1}^{21}\) (m). The reconstructed displacement was compared with the FEM-simulated one and the results are shown in Figures 17-20. The fitting accuracy of the PI_DeepONet for the plate is shown in Table 7. The results indicate that the reconstructed displacement generally matches the FEM-simulated displacement, which demonstrates the broad applicability of PI_DeepONet. Figure 17: Reconstruction results of the displacement for the plate with one side clamped under different loading conditions: (**a**) reconstruction results under loading condition \(T_{P}\), (**b**) reconstruction results under loading condition \(T_{U}\), (**c**) reconstruction results under loading condition \(T_{S}\), and (**d**) reconstruction results under loading condition \(T_{T}\). Figure 18: Reconstruction results of the displacement for the plate with two sides simply supported under different loading conditions: (**a**) reconstruction results under loading condition \(T_{P}\), (**b**) reconstruction results under loading condition \(T_{U}\), (**c**) reconstruction results under loading condition \(T_{S}\), and (**d**) reconstruction results under loading condition \(T_{T}\).