## 3 Validation Tests
### Forward Problems  
Forward problems in solid mechanics involve predicting the mechanical response of a material subjected to boundary conditions, loads, and initial conditions, given its material properties. These problems are often used to simulate the behavior of materials or structures under different boundary conditions in order to gain a better understanding of their response, such as stress, strain, deformation, and displacement. Generally, these problems are typically solved using numerical techniques such as FEM and FDM, solving the equations of motions of the problem. Such techniques allow for the accurate application of boundary conditions, material properties, and other physical constraints. Here, we try to solve this forward problem in solid mechanics using the PINN approach. The governing equations and the boundary condition equations for the plate problem are shown in Figure 2, and they can be expounded as follows:  
Governing equations:  
\[\left\{ \begin{array}{l}\frac{E}{1-v^{2}}\left(\frac{\partial^{2}u_{y}}{ \partial x^{2}}+\frac{1+\mu}{2}\frac{\partial^{2}u_{y}}{\partial x\partial y} +\frac{1-\mu}{2}\frac{\partial^{2}u_{y}}{\partial y^{2}}\right)+f_{x}=0\\ \frac{E}{1-v^{2}}\left(\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{1+\mu}{ 2}\frac{\partial^{2}u_{x}}{\partial x\partial y}+\frac{1-\mu}{2}\frac{ \partial^{2}u_{y}}{\partial x^{2}}\right)+f_{y}=0\end{array} \right. \tag{9}\]  
Boundary conditions equations:  
\[\left\{ \begin{array}{l}\frac{E}{1-v^{2}}\left[l\left(\frac{\partial u_{y}}{ \partial x}+\mu\frac{\partial u_{y}}{\partial y}\right)+m\frac{1-\mu}{2}\left( \frac{\partial u_{y}}{\partial x}+\frac{\partial u_{y}}{\partial y}\right) \right]=\overline{f}_{x}\\ \frac{E}{1-v^{2}}\left[m\left(\frac{\partial u_{y}}{\partial y}+\mu\frac{ \partial u_{y}}{\partial x}\right)+l\frac{1-\mu}{2}\left(\frac{\partial u_{y} }{\partial x}+\frac{\partial u_{y}}{\partial y}\right)\right]=\overline{f}_{y }\end{array} \right. \tag{10}\]  
where  
\[l=cos(\mathbf{n},\mathbf{x}) \tag{11}\]  
\[m=cos(\mathbf{n},\mathbf{y}) \tag{12}\]  
In total, 1500 sample points were randomly generated across the whole domain, including both the interior computational domain and boundary computational domain, with about 20% of the sample points uniformly distributed at the boundary. Additionally, a fully connected neural network (FCN) was applied to predict \(u_{x}\) and \(u_{y}\), with eight hidden layers and 64 neurons per hidden layer, and tanh was here employed as the activation function throughout this study. The FEM-calculated results, achieved using the commercial software ABAQUS with a total mesh of 2400 elements (CPS4R in ABAQUS notation, shown in Figure 3), were used here to express the exact solution. Figure 2: Diagram of an open-holed square plate under a uniaxial load. The training loss versus epoch plot (Figure 4) illustrates the convergence behavior of the PINN model over 10,000 training iterations. The plot shows a steady decrease in the training loss over the course of the training process, indicating that the model effectively learned the underlying physics and reduced the discrepancy between the predicted and true values. The training loss decreased rapidly in the initial epochs and then gradually stabilized to a low value, suggesting that further training may not result in significant improvements. Overall, the plot demonstrates that the PINN model is capable of efficiently learning complex physical phenomena with a relatively small number of training epochs. The displacement fields (\(u_{x}\) and \(u_{y}\)) calculated via the PINN approach are presented in Figure 5 and show satisfactory agreement with the results derived from the FEM. In order to more intuitively illustrate the differences between the displacement fields obtained by the FEM and PINN, an absolute error map is presented in Figure 6. As shown here, the PINN approach successfully predicted the displacements, while a maximum absolute error of only \(-0.00401\) was found.