## 3 Validation Tests
### Inverse Problems
It can be seen that deviation increased with increases in noise scale. In general, the values of the converged Poisson's ratio \(v\), as derived from the PINN approach at different noise scales, were almost identical to the exact solution, which indicates the good performance of the PINN when used for inversion. When the PINN was applied to the \(\epsilon_{1}\) noise scale, it required 1221 iterations to finish training for the 2D inversion problem, while applying the same approach to the \(\epsilon_{2}\) and \(\epsilon_{2}\) noise scales required about2523 and 3851 iterations, respectively. This indicates that noise scale has a strong effect on inversion efficiency. We now seek to further verify the applicability of the PINN to inverse problems. Accordingly, a more challenging task designed to reveal the unknown physics law at play in displacement observations is presented here; said law is assumed to follow an exponential form:  
\[v=v(x)=\alpha e^{x} \tag{13}\]  
where \(x\) donates the horizontal coordinate ranging from 0 to 1 and \(\alpha\) donates the coefficient, set at the constant of 1/3 in this problem. Exact displacement observations were generated using the FEM (using the mesh scheme shown in Figure 3) via Equation (13), implemented on ABAQUS via user-defined materials (UMATs). The goal is to predict the evolution law of \(v\) based on the exact displacement observations and the assumption that \(v\) is simply a function of \(x\), whereby the specific form of \(v(x)\) is a priori unknown. This task was challenging compared to the previous one of simply identifying an unknown parameter during training. In order to approximate the exact evolutionary law of \(v(x)\), an array \(v_{n}\) with \(n\) elements was established, with the Poisson's ratio set in the horizontal direction uniformly divided into \(n\) layers. A schematic diagram of \(v_{n}\) is shown in Figure 10, i.e., \(v_{i}\) represents the Poisson's ratio of \(i\)th layer. The values of \(n\), representing the number of unknown parameters requiring identification, are set as 5, 10, and 15 during the inversion process, respectively. The details and parameters of the training follow those used in the previous study (Table 1). Similarly, \(n\) parameters were optimized together with the weights and biases in the neural network during the training process. The converged \(v_{n}\) after inversion is shown in Figure 10. The circle, square, and triangle marks represent the results derived from the PINN with the values of \(n\) set as 5, 10, and 15, respectively. The black line represents the exact solution obtained from Equation (13). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Noise Scale** & **Identified \(v\)** & **Deviation** & **Number of Iterations** \\ \hline \(e_{1}\) & 0.29802 & 0.00198 & 1221 \\ \(e_{2}\) & 0.30758 & 0.00758 & 2523 \\ \(e_{3}\) & 0.32636 & 0.02636 & 3851 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of different noise scales for inversion. Figure 9: Training samples for \(u\). Figure 11a depicts the loss versus epoch curve (\(n=10\)) for a PINN model used to approximate \(v(x)\). The curve demonstrates the convergence of the model, indicating that the model learned to fit the training data and approximate \(v(x)\) accurately. The gradual decrease in loss with increasing epochs suggests that the model was learning and adapting to the problem over time and eventually converged to a stable solution. In general, the converged \(v_{n}\) derived from the PINN showed good agreement with the exact solution (shown in Figure 11b), illustrating the capacity of the PINN when used to determine unknown physics laws from a complex non-linear system. As observable in Figure 10, the PINN approach successfully predicted the evolutionary law of the Poisson's ratio as given by Equation (13), while a maximum discrepancy of only 0.131 was found when \(n=15\). It can be inferred that \(n=10\) is the best choice when seeking to solve this problem using a PINN.