## 3 Validation Tests
### Inverse Problems
This is because, when the number of unknown parameters is too small, the \(v_{n}\) usually fails to directly reflect the true evolution law. However, when too many parameters are required to be identified at the same time, the calculation accuracy suffers, which may limit the success of identifying the true physics laws. To further test the ability of PINNs to handle inverse problems, we present an example of using a PINN to invert a non-homogeneous modulus field along the \(y\) direction (shown in Figure 12). The model used here was the same as the one used to invert for an unknown  
Figure 11: (**a**) Training loss versus epoch (\(n=10\)); (**b**) Converged \(v_{n}\) (\(n=5,\ 10\), and \(15\)) obtained from the PINN in the \(x\) direction. Figure 10: Schematic diagram for \(v_{n}\) uniformly distributed in the \(x\) direction. Poisson ratio distribution discussed earlier. We assume that the elastic modulus follows the distribution:  
\[E=E(y)=4(y-0.5)^{2}+1 \tag{14}\]  
As seen from Equation (14), the modulus followed a quadratic distribution in the \(y\) direction. To solve this problem, we adopted a method of solving for the Poisson's ratio, which is not discussed here for brevity. The converged \(E_{n}\) after inversion is shown in Figure 12. 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 (14). The convergence curve of the training for this model is similar to that of the Poisson example and is thus not presented here. From Figure 13, it is apparent that the three inversion results showed good agreement with the true result and were all capable of reflecting the distribution pattern of the true modulus. On the whole, it can be observed that when \(n=10\), the degree of agreement between the inversion results was higher, which is highly consistent with the conclusion obtained from the inversion of the Poisson ratio distribution mentioned earlier. This example serves as further evidence of the powerful ability of PINNs in dealing with inverse problems. Figure 12: Schematic diagram for \(E_{n}\) uniformly distributed in the \(y\) direction. Figure 13: Converged \(E_{n}(n=5,\ 10,\) and 15) obtained from the PINN in the \(y\) direction. Table 2 presents the computation time required by the PINN and FEM methods for the three examples mentioned earlier, namely, the inversion of a single Poisson's ratio, the inversion of a Poisson ratio distribution with an unknown form (\(n=10\)), and the inversion of a modulus field with an unknown form (\(n=10\)). For simplicity, we will label these three examples as 1, 2, and 3, respectively. In this study, we utilized the finite element model updating method (FEMU) to perform parameter inversion for these examples, based on the approach described in [42; 43]. Notably, the differences in accuracy between the FEMU and PINN for the inverse problem were insignificant; therefore, we focused on comparing their computational efficiency. The experiments were conducted on a computer equipped with an Intel i7 processor (8-core) and 16 GB of RAM, and computations were accelerated by a GeForce RTX 2070 GPU. Our results demonstrate that the PINN method achieved a remarkable reduction in computational time compared to the FEM, especially in scenarios that required the inversion of multiple parameters. As shown in Table 2, the inversion time with the PINN was up to 90% lower than that with the FEM. For instance, the inversion time for Examples 2 and 3 was reduced from 625 s and 674 s with the FEM to 63 s and 66 s with the PINN, respectively. This highlights the innovative potential of the PINN method in solid mechanics and emphasizes its capability for efficient and accurate inverse analysis. By minimizing computational time, the PINN method provides researchers with the opportunity to expedite their investigations and make informed decisions promptly.