## 2 Materials and Methods
### Direct, Inverse Neural Network-Based Material Parameter Identification Process  
In combination with a numerical model being able to adequately reproduce reality, the MPI has a major influence on structure modeling and sustainability assessment. In addition to the trial-and-error approach, the MPI process consists of solving an inverse identification problem to identify or estimate the system parameters based on the measured system response [38]. For solving this identification method, two general methods are distinguished (see Figure 2). In the most commonly used iterative optimization-based method, an error function is defined as the difference between experimental measurements and parameterized model outputs. This error is usually minimized using an optimization algorithm (e.g., gradient-based or genetic algorithm [39; 40]). Since no standardized procedure or terminology exists, this method is called, e.g., by [PERSON] [41], the forward (classical) mode. With the expanding application of artificial intelligence (AI), a hybrid ML-based method emerged based on the forward mode, wherein time-consuming numerical simulations are replaced by surrogate models such as neural networks, and then the error measure is minimized using an optimization algorithm [42]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **V1** & **V2** & **V3** & **V4** \\ \hline Velocity (\(\frac{mm}{s}\)) & 0.0166 & 0.1666 & 16.666 & 1666.6 \\ Strain Rate (\(\frac{1}{s}\)) & \(1\times 10^{-4}\) & \(1\times 10^{-3}\) & \(1\times 10^{-1}\) & \(1\cdot 10^{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test velocities and resulting strain rates of the virtual dataset. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & **Sampling** & **Sampling** & **Training Set** & **Validation** & **Complete** \\  & **Method** & **Type** & **Size** & **Set Size** & **Set Size** \\ \hline
1 & LHS & individually & 266 & 114 & 380 \\
2 & LHS & individually & 525 & 225 & 750 \\
3 & LHS & individually & 1050 & 450 & 1500 \\
4 & LHS & individually & 2100 & 900 & 3000 \\
5 & LHS & complete & 266 & 114 & 380 \\
6 & LHS & complete & 525 & 225 & 750 \\
7 & LHS & complete & 1050 & 450 & 1500 \\
8 & LHS & complete & 2100 & 900 & 3000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Datasets used for the comparative study. Figure 2: Different methods for material parameter identification. In the second main approach, the existence of an inverse relationship between output and input was assumed, and by effectively inverting the material model, a surrogate model (e.g., NN) was trained using material responses from finite element analysis as input. Afterwards, the model can be applied to directly predict the model (material) parameters and is therefore called an direct inverse or inverse mode [6,41]. Here, the main advantage is the computationally cost-efficient prediction of the material parameters for repeated MPI, since the generated knowledge is stored in the parameters of the surrogate model and new numerical simulations are not needed [7]. As for the classical forward mode, no standardized procedure for the direct inverse approach exists, and many different methods can be implemented into the process. In the following, the specific Python-based implementation of the direct inverse method used in this work is essentially described (see Figure 3), and reference is made to [7] for a detailed explanation. In the first step, a material parameter range had to be defined, which ascertained sufficient coverage of the material characteristics. By applying appropriate sampling methods such as LHS, MP sets were generated, and the corresponding material card input curves/tables were generated using the equations presented in Section 2.2. In this work, the resulting system responses comprise the respective force-displacement curves (FDC) and the plastic Poisson's ratio-equivalent plastic strain curves (PEC) of the numerical simulations.