## 2 Problem Description
For the line (see Figure 2) at \(y=A(A\in\;\left[0,l^{\prime}\right])\) on a cross-section parallel to the plane \(xOz\), the strain-displacement relation (7) of the plate degenerates into the same form as that of the beam. Therefore, we only need the strain along the \(x\) coordinate direction to reconstruct the vertical displacement of the cross-section at \(y=A\). The boundary restrictions of the beam or plate are categorized as free, simply supported, clamped, and elastically supported. We neglect the case of elastically supported boundary restrictions and consider only the geometric boundary conditions specifying \(\mathbf{w}\) or \(\frac{dw}{dx}\). The corresponding boundary conditions for each restriction are presented in Table 1. \begin{table}
\begin{tabular}{l l} \hline \hline
**Restriction** & **Boundary Conditions** \\ \hline Free & \(\mathbf{w}\) or \(\frac{dw}{dx}\) are not specified \\ Simply supported & \(\mathbf{w}=0\) \\ Clamped & \(\mathbf{w}=0\), \(\frac{dw}{dx}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Boundary conditions for different restrictions. Figure 2: Cartesian coordinate system used for the plate analysis. The strain-displacement relations, together with the boundary conditions of the structure, are referred to as GDEs in this paper. In Table 1, the boundary conditions can be categorized into the Dirichlet boundary condition, specifying \(\mathbf{w}\), and the Neumann boundary condition, specifying \(\frac{d\mathbf{w}}{dx}\). To facilitate the training of the network, the Dirichlet boundary conditions at all boundaries are combined into one equation. As such, the general form of the GDEs can be expressed as follows:  
\[\left\{ \begin{array}{ll}\mathcal{P}(x,\mathbf{\varepsilon},\mathbf{w})=0,&x\in\Omega \\ \mathcal{N}_{i}(x,\mathbf{\varepsilon},\mathbf{w})=0,&x\in\partial\Omega_{i},i=1, \ldots,H\\ \mathcal{D}(x,\mathbf{\varepsilon},\mathbf{w})=0,&x\in\partial\Omega_{1}+\partial \Omega_{2}+\ldots+\partial\Omega_{J}\end{array} \right. \tag{8}\]  
in which \(\mathcal{P}\) is the general differential operator that defines the strain-displacement relation, \(\left\{\mathcal{N}_{i}\right\}_{i=1}^{H}\) and \(\mathcal{D}\) denote the Neumann boundary conditions and Dirichlet boundary conditions, respectively, \(H\) and \(J\) denote the numbers of Neumann boundary conditions and Dirichlet boundary conditions, respectively, and \(\Omega\subset\mathbb{R}\) and \(\partial\Omega\) denote the geometric domain and the boundary region of the structure, respectively. In summary, solving the GDEs under different loading conditions with the full-field strain, as described in Equation (8), can reconstruct the structural displacement. However, the arrangement of strain measuring points is constrained in practice, resulting in an unknown strain distribution between adjacent measuring points. Therefore, the numerical solution of GDEs is confronted with inherent challenges. Indeed, the process of reconstructing displacement from strain under different loading conditions essentially constitutes an operator regression problem. To effectively perform the operator regression while removing the dependence on full-field strain, a physics-informed DeepONet based on the GDEs is constructed in this paper.