## 3 Methodology  
### Physics-Informed DeepONet  
DeepONet was designed to acquire abstract nonlinear operator mapping functions between Banach spaces of infinite dimensions [26]. Compared to fully connected neural networks, DeepONet significantly decreases the generalization error while guaranteeing a smaller approximation error. By introducing an efficient regular mechanism, PI_DeepONet biases the output of the DeepONet model to ensure physical consistency. Here, we present a brief overview of the concept and architecture of the PI_DeepONet. Let \(\mathcal{U}\) and \(\mathcal{W}\) be two separate branch spaces. Our purpose is to learn the solution operator \(G\) that maps strain function \(\mathbf{\varepsilon}\in\mathcal{U}\) to displacement function \(\mathbf{w}\in\mathcal{W}\), which is defined as follows:  
\[G(\mathbf{\varepsilon})=\mathbf{w} \tag{9}\]  
The solution operator \(G\) is represented by the DeepONet \(G_{\theta}\), in which \(\theta\) denotes all trainable parameters of the DeepONet. The architecture of the DeepONet is shown in Figure 3. The DeepONet consists of two separate neural networks, called branch net and trunk net, respectively. The branch net receives the strain function \(\mathbf{\varepsilon}\) as its input. The strain values at \(m\) locations are used for the expression of the strain function \(\mathbf{\varepsilon}\) as follows:  
\[\mathbf{\varepsilon}=\left[\mathbf{\varepsilon}(x^{(1)}),\mathbf{\varepsilon}(x^{(2)}), \cdots,\mathbf{\varepsilon}(x^{(m)})\right] \tag{10}\]  
After the \(\mathbf{\varepsilon}\) is fed into the branch net, a feature embedding \(\left[b_{1},b_{2},\ldots,b_{r}\right]^{T}\in\mathbb{R}^{r}\) is returned as the output. The trunk net receives the coordinate \(x\) as an input, which is one-dimensional in this paper. The feature embedding \(\left[t_{1},t_{2},\ldots,t_{r}\right]^{T}\in\mathbb{R}^{r}\) is the output of the trunk network. It is worth noting that the output layers of both the trunk net and the branch net consist of the same number of neurons. The number of neurons in the input layer of the trunk net and branch net can be determined based on the dimension of the inputs \(x,\epsilon\). The final output of DeepONet is obtained by computing the inner product of the feature embeddings of the trunk net and branch net. Thus, after inputting the strain function \(\epsilon\) and the coordinate \(x\), DeepONet returns the displacement prediction at the \(x\) coordinate as follows:  
\[G_{\theta}(\epsilon)(x)=\sum_{k}^{r}b_{k}\Big{(}\epsilon\Big{(}x^{(1)}), \epsilon\Big{(}x^{(2)}),\cdots,\epsilon\Big{(}x^{(m)}\Big{)}\Big{)}t_{k}(x)= \sum_{k=1}^{r}b_{k}t_{k} \tag{11}\]  
Classical networks, such as the fully connected neural network (FNN), convolutional neural network (CNN), or recurrent neural network (RNN), can be chosen as the trunk net and branch net according to the input structure. In this paper, an improved fully connected network architecture, proposed by [PERSON] et al. [39], is employed. In contrast to the FNN, the improved architecture introduces two transformer networks that map the input variables to a high-dimensional feature space. The improved architecture explicitly considers multiplicative interactions among input dimensions and enhances the hidden states by introducing residual connections. The forward-propagation of the improved architecture can be expressed as follows:  
\[U=\sigma\Big{(}XW^{1}+b^{1}\Big{)},V=\sigma\Big{(}XW^{2}+b^{2}\Big{)} \tag{12}\]  
\[Z^{(1)}=\sigma\Big{(}XW^{z,1}+b^{z,1}\Big{)} \tag{13}\]  
\[H^{(k+1)}=\Big{(}1-Z^{(k)}\Big{)}\odot U+Z^{(k)}\odot V,\ k=1,\ldots,L \tag{14}\]  
\[Z^{(k)}=\sigma\Big{(}H^{(k)}W^{z,k}+b^{z,k}\Big{)},\ k=2,\ldots,L \tag{15}\]  
\[f_{\theta}(x)=H^{(L+1)}W+b \tag{16}\]  
In Equations (12)-(16), \(\odot\) represents the inner product matrix multiplication, \(\sigma\) represents the activation function, \(X\) represents the input to the network, \(W\) and \(b\) represent the weights and bias of each layer of the network, and \(f_{\theta}(x)\) represents the final output of the network.