## 4 Results
### Beam
#### 4.1.1 Displacement Reconstruction for Rectangular Section Beam  
This section focuses on reconstructing the displacement for the rectangular section beam with different restrictions. The restrictions of the beam used for the displacement reconstruction were clamped at one end, simply supported-simply supported, clamped-simply supported, and clamped-clamped. Figure 5 shows the geometric model of the beam with one end clamped. The thickness of the beam is 0.03 m. To reconstruct the vertical displacement, 10 strain measuring points were arranged on the surface of the beam, as shown in Figure 5. The \(x\) coordinates of the strain measuring points are \(\left\{0.02+0.1(i-1)\right\}_{i=1}^{10}\) (m). To reconstruct the displacement of the beam, a corresponding PI_DeepONet for different restrictions was built and trained. Here, we present the specific implementation of the PI_DeepONet using the beam with one end clamped as the example. Based on the arrangement of the strain measuring points, the numbers of neurons in the input layer of the branch net and the trunk net were designed to be 10 and 1, respectively. Both trunk net and branch net have three hidden layers, with 1000 neurons per layer. Combining the strain-displacement relation and boundary conditions, the GDEs of the beam with one end clamped can be expressed as follows:  
\[\left\{ \begin{aligned} \mathbf{\epsilon_{x}}(x)+0.5t\frac{d^{2} \mathbf{w}(x)}{dx^{2}}=0\\ \frac{d\mathbf{w}(x)}{dx}=0,\ x=0\\ \mathbf{w}(x)=0,\ x=0\end{aligned} \right. \tag{35}\]  
According to the GDEs of the beam, each individual physics-driven loss and the aggregate loss of the PI_DeepONet can be defined as follows:  
\[L_{p}(\theta)=\frac{1}{Nm}{\sum_{i=1}^{N}}{\sum_{j=1}^{m}}{\left|\mathbf{ \epsilon_{i}}\left(x^{(j)}\right)+0.5t\frac{d^{2}G_{\theta}(\mathbf{\epsilon_{i} })\left(x^{(j)}\right)}{dx^{2}}\right|^{2}} \tag{36}\]  
\[L_{B_{1}}(\theta)=\frac{1}{NQ_{1}}{\sum_{i=1}^{N}}{\sum_{j=1}^{Q_{1}}}{\left| \frac{dG_{\theta}\left(\mathbf{\epsilon_{i}}^{r}\right)\left(x_{B_{1}}^{(j)} \right)}{dx}-0\right|^{2}},\ x_{B_{1}}^{(j)}=0 \tag{37}\]  
\[L_{B_{2}}(\theta)=\frac{1}{NQ_{2}}{\sum_{i=1}^{N}}{\sum_{j=1}^{Q_{2}}}{\left| G_{\theta}\left(\mathbf{\epsilon_{i}}^{r}\right)\left(x_{B_{2}}^{(j)}\right)-0 \right|^{2}},\ x_{B_{2}}^{(j)}=0 \tag{38}\]  
Figure 5: The geometric model of the beam with one end clamped. \[L(\theta)=\lambda_{p}L_{p}(\theta)+\lambda_{1}L_{B_{1}}(\theta)+\lambda_{2}L_{B_{2} }(\theta) \tag{39}\]  
in which \(\left\{x^{(j)}\right\}_{i=1}^{m}\) are coordinates of the strain measuring points, and \(\left(\lambda_{p},\lambda_{1},\lambda_{2}\right)\) are the weights of the losses. To construct the training datasets for each individual loss, 1000 random strain functions \(\epsilon^{r}\) were modeled using GRF. After the training datasets had been constructed, 80,000 iterations of gradient descent were performed to train the PL_DeepONet. The convergence process of each physics-driven loss is shown in Figure 6. The PL_DeepONet converged to the solution operator of the GDEs by minimizing each individual physics-driven loss of GDEs to less than \(10^{-6}\). Similarly, for the beam with other restrictions, a corresponding PL_DeepONet was built and trained to reconstruct the displacement. For the beam with the restrictions of clamped-clamped, the Neumann boundary conditions at both ends of the beam need to be fitted simultaneously by the PL_DeepONet, resulting in complex loss terms. In this scenario, to mitigate the imbalance between the gradients of each loss throughout the training process, Algorithm 1 was used to perform gradient descent iterations. Recording the updated weights every 100 iterations, the convergent evolutions of the weights when training the PL_DeepONet are summarized in Figure 7.  
Figure 6: Convergence process of each loss when training the PI_DeepONet for the beam with one end clamped.