## 4 Results
### Beam
#### 4.1.1 Displacement Reconstruction for Rectangular Section Beam
Figure 7: Convergent evolution of the weight of the each physics-driven loss when training the PL_DeepONet for the beam with two ends clamped using Algorithm 1: \(\lambda_{p}\) denotes the weight of loss of strain–displacement relation, \(\lambda_{1}\) and \(\lambda_{2}\) denote the weights of the loss of Neumann boundary conditions at point \(x=0\) m and \(x=1\) m, respectively, and \(\lambda_{3}\) denotes the weight of the loss of Dirichlet boundary conditions. For the beam, it is also important to reconstruct rotating angle \(\beta\) of the cross-section (see Figure 1). In the case of small displacements, rotating angle \(\beta\) can be calculated by taking the first-order derivative of the displacement with respect to coordinate \(x\). The displacement is the output of the PI_DeepONet, and coordinate \(x\) is one of the inputs. Thus, after inputting strain function \(\epsilon_{i}\) and coordinate \(x_{j}\), rotating angle \(\beta\) at coordinate \(x_{j}\) can be calculated via PI_DeepONet as follows:  
\[\beta=\frac{dG_{\theta}(\epsilon_{i})(x_{j})}{dx} \tag{40}\]  
In order to evaluate the performance of the PI_DeepONet, the reconstructed displacement and rotating angle of the beam under three loading conditions were compared to those simulated by FEM. The magnitude and position of each load applied to the beam are shown in Table 2, in which \(T_{P}\) denotes a point force applied at a certain location, \(T_{U}\) denotes uniform pressure applied to the surface of the structure, and \(T_{S}\) denotes two point forces acting in opposite directions at different locations. The reconstruction results of the PI_DeepONet are shown in Figures 8-11, in which the \(x\) coordinates of displacement and the rotating angle measurement points are \(\{0.05(i-1)\}_{i=1}^{21}\) (m). The fitting accuracy, indicating the degree of agreement between the reconstructed terms and the simulated terms, was used to evaluate the performance of PI_DeepONet. The fitting accuracies of PI_DeepONet for the beam with different restrictions are shown in \[L_{B_{1}}(\theta)=\frac{1}{NQ_{1}}{\sum_{i=1}^{N}}{\sum_{j=1}^{Q_{1}}}{\left| \frac{dG_{\theta}\left(\mathbf{\epsilon_{i}}^{r}\right)\left(x_{B_{1}}^{(j)} \right)}{dx}-0\right|^{2}},\ x_{B_{1}}^{(j)}=0 \tag{37}\]0 The results show that the reconstructed displacement and rotating angle are basically consistent with the simulated results, and the fitting accuracies are all above 0.99. The results verify the excellent performance of the PI_DeepONet on the reconstruction of displacement and rotating angle for beam. Furthermore, no strain measuring points are positioned at the boundaries (\(x=0\) m and \(x=1\) m) of the beam, demonstrating that our method is not reliant on the full-field strain. \[L_{B_{1}}(\theta)=\frac{1}{NQ_{1}}{\sum_{i=1}^{N}}{\sum_{j=1}^{Q_{1}}}{\left| \frac{dG_{\theta}\left(\mathbf{\epsilon_{i}}^{r}\right)\left(x_{B_{1}}^{(j)} \right)}{dx}-0\right|^{2}},\ x_{B_{1}}^{(j)}=0 \tag{37}\]1 Reconstruction results of the displacement and rotating angle of the beam with the restrictions of clamped–clamped under different loading conditions. \[L_{B_{1}}(\theta)=\frac{1}{NQ_{1}}{\sum_{i=1}^{N}}{\sum_{j=1}^{Q_{1}}}{\left| \frac{dG_{\theta}\left(\mathbf{\epsilon_{i}}^{r}\right)\left(x_{B_{1}}^{(j)} \right)}{dx}-0\right|^{2}},\ x_{B_{1}}^{(j)}=0 \tag{37}\]2 Reconstruction results of the displacement and rotating angle of the beam with the restrictions of clamped–simply supported under different loading conditions.