## 4 Discussions
### Error Quantization Format of PartsNet  
Error analysis is a crucial component in quantitatively describing the accuracy of numerical computations. In this study, we proposed the error quantification format for parameter calibration. Errors can be broadly categorized into systematic errors and random errors. Systematic errors can be understood as the limited expressive power of numerical models and the constrained optimization ability of algorithms, including deep learning. Random errors arise from the varying degrees of numerical discrepancies observed in experiments and calculations. \[E=E_{systematic}+E_{stochastic} \tag{27}\]  
Here,  
\[\left\{ \begin{aligned} E_{systematic}=e_{modding}+e_{optimizing}\\ E_{stochastic}=e_{experiment}\end{aligned} \right.\]  
Indeed, the analysis of random errors in mechanical engineering can be complex. Common sources of such errors include limitations in sensor accuracy and unavoidable environmental noise, leading to measurement errors. In this study, we currently do not focus on quantifying these errors stemming from inherent priors. Instead, our main emphasis is on quantifying systematic errors. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **MSE** & **MAE** & **RMSE** & **MedAE** & **Qloss** & **R\({}^{2}\)** & **Time** \\ \hline RNN & 0.024 & 0.113 & 0.155 & 0.071 & 0.059 & 0.548 & 57.010 \\ GRU & 0.006 & 0.052 & 0.080 & 0.037 & 0.026 & 0.881 & 213.409 \\ RNN+A & 0.011 & 0.067 & 0.106 & 0.039 & 0.030 & 0.791 & 65.679 \\ GRU+A & 0.006 & 0.049 & 0.076 & 0.031 & 0.023 & 0.892 & 232.825 \\ RNN+TransE & 0.020 & 0.100 & 0.141 & 0.062 & 0.044 & 0.629 & 102.360 \\ GRU+TransE & 0.005 & 0.050 & 0.072 & 0.035 & 0.022 & 0.902 & 202.654 \\ RNN+STE & 0.019 & 0.098 & 0.138 & 0.068 & 0.043 & 0.643 & **43.205** \\
**PartsNet** & **0.0009** & **0.019** & **0.031** & **0.011** & **0.008** & **0.913** & 159.586 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Ablation experimental data of PartsNet in triaxial test. Systematic error \(E_{systematic}\) is primarily composed of \(e_{modeling}\), which originates from the constitutive relationship of the model. The quantitative constitutive relationship serves as a dimensional reduction representation of the real system, and the information lost during this process contributes to the error. Systematic error represents the upper limit of all objective limitations of the parameter calibration algorithm. Without loss of generality, this study employs a third-order Taylor expansion to calculate \(E_{systematic}\) and utilizes data-oriented numerical differentiation methods to compute gradients. The novelty of this approach lies in circumventing the issue of dimension reduction caused by deterministic constitutive relationships. Let \(X_{N}\) be an m-dimensional orthogonal sequence with \(N\) samples. We input this sequence into the trained PartsNet and use the predicted parameters to perform a simulation, resulting in another set of sequence data \(Y_{N}\) with the same shape. \[E_{systematic}=\left|\frac{1}{m}\left[\sum_{i=1}^{m}(x_{i}-y_{i})-\sum_{i=1}^{ m}\sum_{j=1}^{order}\frac{\varepsilon^{j}\cdot(adaptive_{difference}(x_{i},\ y_{i}))}{j!}\right]-e_{optimizing}\right| \tag{28}\]  
The \"order\" parameter was set to 3 in this paper, and \(\varepsilon\) represents the expansion position, which we set to \(10^{-7}\). This study introduces an adaptive difference algorithm to calculate the numerical derivatives of the discrete sequence. At each internal data point, the algorithm selects the smallest spacing as the step size based on the distances between neighboring data points and uses the central difference scheme to compute the derivative value. For boundary points, forward and backward difference schemes are still used. This method adjusts the step size based on the local characteristics of the data points, improving the accuracy and stability of numerical differentiation.