## 1 Introduction
A coupled neural network framework with skip connections was proposed to predict unknown parameters and the effectiveness and applicability of the proposed methods was verified by using real experimental results. In the field of geophysics, [PERSON] et al. [16] proposed a new approach to the solution of wave propagation and full waveform inversions (FWIs) based on a PINN approach. In his work, the PINN could seamlessly handle physical constraints and absorb boundary conditions relevant to geophysical applications. [PERSON] et al. [18] presented a PINN to solve the problem of identifying and characterizing a surface-breaking crack in a metal plate, which showed a promising deep neural network model for ill-posed inverse problems. [PERSON] et al. [22] proposed a PINN framework that could enable the seamless synthesis of non-invasive in vivo measurement techniques and computational flow dynamics models derived from first physical principles, and the effectiveness of the proposed techniques was illustrated by a series of realistic clinical examples. More importantly, this work was the first time PINN was applied to solve conservation laws in graph topologies. PINN is a type of artificial neural network that is trained using both labeled and unlabeled data combined with prior physical insights as constraints [23; 24]. It can solve partial differential equations (PDEs) by explicitly embedding physical information (boundary conditions, initial conditions, physical equations, etc.) into loss functions. Incorporating physical insights in this way can restrict the range of models that the algorithm can learn, resulting in more generalizable and interpretable outcomes. By using prior physical insights as constraints, it can not only reduce the dependence of the model on a large training dataset, but also provide physical constraints for the model, thereby improving its prediction accuracy and generalization capacity. Furthermore, it can even produce highly accurate physical simulations in a fraction of the time and cost of traditional numerical simulations [25; 26]. In general, in addition to the increased accuracy, PINNs have several advantages over traditional deep learning and numerical methods. First, they can reduce the computational overhead associated with deep learning since the physics-based equations are already known. Second, fewer training samples are required compared to traditional deep learning models. Finally, PINNs can provide physical insight into the underlying system, thus allowing researchers to better understand the parameters or phenomena that are driving the system's behavior. [PERSON] et al. [27] demonstrated that PINNs can learn complex non-linearities and boundary conditions from data, while incorporating physical constraints, and require much fewer training points than traditional methods to achieve high accuracy. [PERSON] et al. [28] proposed a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions to construct PINN models for surrogate modeling and uncertainty quantification tasks in PDE systems. These models are trained without labeled data, providing comparable predictive responses to data-driven models while obeying the constraints of the problem. Despite their wide use in various scientific computing domains, few studies on PINNs have been reported in solid mechanics, and we have seen a particular dearth in problems involving non-uniform deformation with obvious stress concentrations in specimens with complex shapes. Recently, various changes have been made to PINNs' structures to improve their applicability to some challenging physics problems, such as turbulent flow prediction [10], multiscale modeling [29], quantum many-body systems [30], and inverse problems [31]. However, it has been proven that a PINN with a simple fully connected network (FCN) set as its neural network structure can address most scientific problems.