## 2 Methodology: PartsNet
### General Data Dimensionality Reduction Network (GDR-Net)  
The sequence data generated by numerical models are taken as input for PartsNet. Sequence data comprise multidimensional feature parameters, which exhibit non-orthogonality and varying weights. Characterizing a system can be accomplished by calibrating only a few parameters with the most substantial weights and richest information. Previous works [31; 32] have mainly relied on principal component analysis (PCA) and its improved algorithms to perform data feature dimensionality reduction. PCA is an unsupervised machine learning approach that may fail to recognize nonlinear connections because of its strict assumptions about the data. This paper proposes a general data dimensionality reduction method based on cross-correlation functions to identify significant features in large datasets. We employ Variational Autoencoders (VAEs) [33] to enhance the feature extraction capability of PartsNet. The sequential data consist of a set of signals. Figure 1: Data-driven or model-driven parameter calibration methods shown in three aspects. **Definition 1**. : _For any square-integrable functions, \(X\in\mathbb{R}^{N\times K}\)and\(\widetilde{X}\in\mathbb{R}^{N\times K}\), \(\gamma(m)\) represents the cross-correlation of \(X\) and \(\widetilde{X}\) at m. \(\gamma(m)\) is subjected to_  
\[\gamma(m)=\text{sigmoid}\left(\frac{1}{NK}\int_{K}\int_{N}X(n)\cdot\widetilde{ X}(n+m)\text{ dndk}\right) \tag{1}\]  
_We use the sigmoid function to normalize the values. For discrete signal functions under investigation, it can be further expressed in the following weak form:_  
\[\gamma(m)=\text{sigmoid}\left(\frac{1}{NK}\sum_{k=1}^{K}\sum_{n=1}^{N}X(n) \cdot\widetilde{X}(n+m)\right) \tag{2}\]  
_Specially, let K equal 2, and consider the cross-correlation of discrete sequential data at \(m=0\). \(\gamma(m)\) is subjected to:_  
\[\gamma(m)=\text{sigmoid}\left(\frac{1}{2N}\sum_{n=1}^{N}\left(X_{1}(n)\cdot \widetilde{X_{1}}(n)+X_{2}(n)\cdot\widetilde{X_{2}}(n)\right)\right) \tag{3}\]  
The significance of this approach and the weak form lies in establishing a mutual correlation between simulation-generated sequence data and experimentally obtained data, facilitating the labeling process. A higher value of \(\gamma(m)\) indicates a higher quality of the simulation-generated data, while a lower value indicates the opposite. The core of GDR-Net is the use of a VAE for the dimensionality reduction in labeled feature data [34], as shown in Figure 2. A VAE is a generative neural network based on variational Bayesian inference, which addresses the overfitting issue commonly found in traditional autoencoders [35]. The encoder of a generative model uses the input data to learn a compressed form, known as latent variables, while the decoder uses this compressed representation to reconstruct the input. The differences between the original data and the reconstructed data is termed as a reconstruction error, which serves as the optimization objective of autoencoders. The VAE introduces a set of random variables, denoted as \(z\), to represent the latent distribution \(p(z)\) of the input data and facilitate the generation of new data. The latent model follows a Gaussian distribution, with its mean and variance generated by a parameterized function output by the encoder. The objective of the VAE is to maximize the Evidence Lower Bound (ELBO), which is a lower bound on the log-likelihood of the generative model. The ELBO is defined as follows:  
\[ \begin{split}\mathcal{L}_{1}(\theta,\varphi;\ x)&= E_{q_{\varphi}(z|x)}[\log p_{\theta}(x|z)]-D_{KL}\big{(}q_{\varphi}(z|x)\|p(z) \big{)}\\ &=E_{q_{\varphi}(z|x)}\bigg{[}\sum_{i=1}^{n}\log p_{\theta}(x_{i} |z)\bigg{]}-\int q_{\varphi}(z|x)\log\frac{q_{\varphi}(z|x)}{p(z)}\,dz\end{split}  \tag{4}\]  
Here, \(q_{\varphi}(z|x)\) represents the variational posterior distribution generated by the encoder, \(p_{\theta}(x|z)\) represents the generative distribution generated by the decoder, and \(p(z)\) represents the prior distribution.