## 1 Introduction
However, the effectiveness of the conventional deep learning method heavily relies on training datasets that include expected mechanical responses [18], and building these training datasets is often challenging and costly. To diminish the reliance of neural networks on paired input-output observations, [PERSON] [19] formalized physics-informed neural networks (PINN) in 2019. In various physics and engineering scenarios, priori knowledge in the form of partial differential equations (PDEs) usually exists between input and output observations (e.g., strain-displacement equations in elastic mechanics, Navier-Stokes equations in fluid mechanics). The PINN incorporates the prior knowledge as physical loss terms into the loss function of the neural network. Guided by physical loss terms, PINN requires fewer or no paired input-output observations to be trained and provides better generalization. The application of PINN has expanded to various fields, including fluid mechanics [20], biomedicine [21], materials science [22], fracture mechanics [23], power systems [24], and scientific machine learning (SciML) [25]. Therefore, PINN is a highly promising technique for addressing the issue of displacement reconstruction. Using PINN to reconstruct structural displacement essentially means solving a solution operator that maps the strain function to the displacement function. The Deep Operator Network (DeepONet) [26] is a novel operator learning architecture motivated by [PERSON]'s (1995) universal approximation theorem [27]. The DeepONet significantly reduces the computational cost of solving operator regression problems and provides better generalization and faster convergence compared to traditional fully connected networks. Drawing inspiration from PINN, physics-informed DeepONet (PI_DeepONet) [28] was proposed to efficiently learn a solution operator of PDEs. The PI_DeepONet is an extension of DeepONet and satisfies the underlying PDEs by incorporating them into the loss function of the DeepONet. The PI_DeepONet has shown good performance in diverse fields, such as the prediction of crack path [29], solving heat conduction equations [30], and the prediction of instability waves in hypersonic boundary layers [31]. In this paper, PI_DeepONet is applied to solve geometric differential equations (GDEs) of beam and plate structures with various restrictions. According to the Euler-Bernoulli beam theory and the Kirchhoff plate theory, the strain-displacement relations and boundary conditions of a structure are represented in the differential form. The strain-displacement relations, together with the boundary conditions of the structure, are called GDEs in this paper. Utilizing the automatic differentiation techniques in deep learning [32; 33], each GDEs equation is formulated as a loss term of the PI_DeepONet. The PI_DeepONet used to regularize the GDEs is trained to be a solution operator of GDEs, and the solution operator of the beam or plate is utilized to directly map the strain function to the displacement function under diverse loading conditions. Compared to traditional methods, such as the finite difference method (FDM) [34] and the finite element method (FEM), PI_DeepONet is a mesh-free approach and could break the curse of dimensionality [33]. Due to the insufficient understanding of the regular mechanism at present, PINN has a tendency to converge to an incorrect solution in some scenarios. The loss function of PINN is a weighted combination of different loss terms, resulting in low training efficiency if the weights are selected inappropriately. Therefore, many methods have been proposed to assign a proper weight to each loss term. [PERSON] et al. [35] proposed two approaches, named annealing and cold start, to tune the weights of the loss terms in the initial or final period. [PERSON] et al. [36] obtained optimal weights by defining the upper and lower bounds for different loss terms. However, these non-adaptive methods cannot guarantee that the assigned weights remain optimal throughout the entire PINN training process.