## 3 Validation Tests
### Inverse Problems  
Inverse problems in solid mechanics aim to determine the unknown parameters of a material by observing some measurements of the system [37,38]. These measurements may be displacement, strain, stress, etc. Inverse problems can be useful in determining the parameters of a material when direct measurement is not possible, such as in material characterization or system identification, which are especially important for the analysis and identification of materials and structures. The parameters to be determined include the Young's modulus, Poisson's ratio, and other physical properties. Generally, inverse problems are much more difficult than forward problems involving the determination of the response for a given set of parameters. The difficulty arises from the fact that the measured response does not contain enough information to uniquely determine all the parameters and these equations for solid mechanics problems are typically non-linear. To guarantee a unique solution, one approach is to apply additional constraints to the input parameters. For instance, in the paper by [PERSON] et al. [39], they employed a deep neural network inverse solution to recover pre-crash impact data of car collisions while enforcing physical constraints on the solution space. Specifically, they included a regularization term that penalized non-physical solutions, such as those with negative values for physical parameters. Additionally, [PERSON] et al. [40] proposed a feed-forward neural network-based variational Bayesian learning method for the forensic analysis of traffic accidents. Their approach combined Bayesian inference and neural networks to assess the uncertainty of predicted solutions and provide a probabilistic evaluation of the results. This could aid in detecting multiple solutions and guaranteeing the uniqueness of the most probable solution. In summary, to ensure the uniqueness of solutions in inverse analyses utilizing PINNs, it is crucial to incorporate physical constraints and regularization techniques into the network's architecture and estimate the uncertainty of the predicted solutions. These techniques can prevent the network from predicting non-physical solutions and guarantee the uniqueness of the most probable solution. The inverse problem that we considered here was to identify the Poisson's ratio \(v\) (exact value: 0.3), which was taken as the unknown parameter in this section. The PINN takes the spatial coordinate \(\mathbf{x}\) as the input and outputs the corresponding value of \(\mathbf{u}\). In order to identify the Poisson's ratio \(v\), we simultaneously optimized the weights and biases of the neural network along with the Poisson's ratio by minimizing the differences between the predicted and measured \(\mathbf{u}\) using the loss function, while using 200 points (marked by circles in Figure 9) randomly sampled from the interior of the computational domain for \(\mathbf{u}\) as the known data. To enforce the physics-based constraints, we also incorporated the boundary conditions and governing equation into the loss function. The boundary conditions and governing equations (the same as those introduced in the previous section) were taken as a priori. Accordingly, the aim here was to estimate the Poisson's ratio \(v\) from the 200 known points of \(\mathbf{u}\) within the interior of the computational domain. In order to examine the applicability of using a PINN to solve inverse problems, Gaussian-distributed noise [41] with a zero mean was generated for all the sampled points. Three different scales of noise were applied to \(\mathbf{u}\), i.e., (1) \(\epsilon_{1}\sim\)\(N(0,\ 0.001)\), (2) \(\epsilon_{2}\sim\)\(N(0,\ 0.005)\), and (3) \(\epsilon_{3}\sim\)\(N(0,\ 0.01)\). The limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimizer with a learning rate of 0.1 was used to train the PINN, with the number of training steps set as 2000. Table 1 illustrates the values of \(v\) that were identified for different scales of noise, with the initial \(v\) used for inversion set as 0.8.