## 3 Methodology
### Implementation of the PI_DeepONet  
In this section, we provide a concise overview of the implementation of the PI_DeepONet described above. The building and training process of all the PI_DeepONet described in this paper are implemented in the JAX framework. To reconstruct the displacement in different scenarios, it is imperative to build a PI_DeepONet with an appropriate architecture. The architecture of the PI_DeepONet needs to be determined based on the arrangement of the strain measuring points. To predict the displacement \(\mathbf{w}\) via strain \(\epsilon\), let the coordinates of surface strain measuring points be \(\left\{x^{(i)}\right\}_{i=1}^{m}\). The numbers of neurons in the input layer of the branch net and the trunk net are equal to the dimensions of coordinate \(x^{(i)}\) and the measured strain \(\epsilon\), respectively. There is no specific requirement for the hidden layer architectures of the trunk net and branch net, but they can be adjusted appropriately to improve the fitting ability of the PI_DeepONet. All the trunk nets and branch nets of the PI_DeepONet built in this paper have three hidden layers, with 1000 neurons per layer. After the architecture of the PI_DeepONet is determined, the physics-driven losses are determined based on the GDEs of the structure. Training the PI_DeepONet necessitates the construction of a training dataset for each physics-driven loss. The trained PI_DeepONet serving as the solution operator should be capable of providing the solution function of the GDEs for arbitrary function inputs. Therefore, function \(\epsilon\), used for training the PI_DeepONet, is not required to be derived from simulation or experimentation. Here, we used mean-zero Gaussian random fields (GRF) to model random strain functions \(\epsilon^{r}\) to construct the corresponding training datasets for each physics-driven loss as follows:  
\[\epsilon^{r}\sim\mathcal{G}(0,k_{l}(x_{1},x_{2})) \tag{26}\]  
with an exponential quadratic covariance kernel \(k_{l}(x_{1},x_{2})=\exp\left(-\big{|}\big{|}x_{1}-x_{2}\big{|}\big{|}_{2}/2l^{2}\right)\) with a length scale parameter \(l>0\). The parameter \(l\) determines the complexity of the \(\epsilon^{r}\), and a larger \(l\) will produce a smoother \(\epsilon^{r}\). The \(\epsilon^{r}\) are represented by discrete values evaluated at the coordinates of the strain measuring points of the structure. We only need quantities of strain, instead of paired strain-displacement observations, to construct the training datasets. Different physics-driven losses correspond to the different training datasets. For instance, the training dataset of \(L_{p}(\theta)\) takes the following form:  
\[\left[ \begin{array}{c}\vdots\\ \epsilon_{i}^{r}\big{(}x^{(1)}\big{)},\epsilon_{i}^{r}\big{(}x^{(2)}\big{)}, \cdots,\epsilon_{i}^{r}\big{(}x^{(m)}\big{)}\\ \epsilon_{i}^{r}\big{(}x^{(1)}\big{)},\epsilon_{i}^{r}\big{(}x^{(2)}\big{)}, \cdots,\epsilon_{i}^{r}\big{(}x^{(m)}\big{)}\\ \vdots\\ \epsilon_{i}^{r}\big{(}x^{(1)}\big{)},\epsilon_{i}^{r}\big{(}x^{(2)}\big{)}, \cdots,\epsilon_{i}^{r}\big{(}x^{(m)}\big{)}\\ \vdots\end{array} \right]\left[ \begin{array}{c}\vdots\\ \epsilon_{i}^{r}\big{(}x^{(1)}\big{)}\\ \epsilon_{i}^{r}\big{(}x^{(2)}\big{)}\\ \vdots\\ \vdots\end{array} \right]\right] \tag{27}\]  
in which \(\{\epsilon_{i}^{r}\}_{i=1}^{N}\) denotes \(N\) separate random strain functions, modeled by GRF. The PI_DeepONet can be trained once the training datasets have been constructed.