# A Novel Reconstruction Method for Irregularly Sampled Observation Sequences for Digital Twin
## 4 Simulation Experiments and Analysis
### Analysis of Model Reconstruction Results Under Different Sampling Numbers  
The dataset for model training and validation consists of 150 irregular observation sequences, including 5 different random sampling sequences with a sampling number of \(N\) under 30 different working conditions. This section compares the reconstruction results of models under sampling numbers of \(N=60\), \(N=40\), and \(N=20\). The noise covariance of the actual observation sequence is set to  
\[\mathbf{R}= \text{diag}\Big{(}R_{1}\quad R_{2}\quad R_{3}\quad R_{4}\Big{)}\] \[= \text{diag}\Big{(}5\times 10^{-4}\quad 1\times 10^{-2}\quad 2 \times 10^{-2}\quad 5\times 10^{-3}\Big{)}\]  
Figures 3 and 4 show the reconstruction results of each model for the normalized values of each observation variable under different sampling numbers. Since RNN-NODE does not consider the impact of control inputs on outputs, its predictions rely solely on the initial latent state, resulting in significant estimation errors. After improvement with NCDE, the model can account for control sequences, enhancing reconstruction accuracy. As shown in Table 4, RNN-NCDE reduces the estimation error by 66.3%, 65.6%, and 53.5% compared to RNN-NODE when the sampling numbers are 60, 40, and 20, respectively. Although RNN-NCDE improves the reconstruction results of RNN-NODE, its performance degrades noticeably under sparse measurements, as evidenced by a near doubling of prediction errors when the sampling number decreases from 60 to 20.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt}} \hline \hline
**Sequence** & **Network Architecture Parameters** \\ \hline \multirow{5}{*}{PRN-NCDE} & (1) MCSA module: number of channels \(d=4\); number of attention heads \(M=4\); fill value \(\tau=1\times 10^{-7}\); \\  & (2) Number of LSTM network neurons: 80; \\  & (3) Number of encoder fully connected layer neurons: 100; \\  & (4) Latent feature dimension \(n=10\); \\  & (5) NCDE module (two-channel feedforward network): [100]-[100]-[100]-[10]. \\ \hline \multirow{5}{*}{RNN-NCDE} & (1) Number of LSTM network neurons: 80; \\  & (2) Number of encoder fully connected layer neurons: 100; \\  & (3) Latent feature dimension \(n=10\); \\  & (4) NCDE module (two-channel feedforward network): [100]-[100]-[100]-[10]. \\ \hline \multirow{5}{*}{RNN-NODE} & (1) Number of LSTM network neurons: 80; \\  & (2) Number of encoder fully connected layer neurons: 100; \\  & (3) Latent feature dimension \(n=10\); \\  & (4) NCDE module:100-100-100-10. \\ \hline \hline \end{tabular}
\end{table}
Table 2: Network architecture parameters of each reconstruction model.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt}} \hline \hline
**Hardware and Software** & **Details** \\ \hline CPU & Intel(R) Core(TM) i9-10900 CPU @ 2.8 GHz \\ GPU & NVIDIA GeForce GTX 1660 SUPER \\ Random Access Memory & 32 GB \\ Matlab Version & 2024a \\ Deep Learning Toolbox & 24.1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Experimental environment configuration.  
Figure 4: _Cont._  
Figure 3: Reconstruction results of gas density and gas temperature under different sampling numbers.  
The proposed PRN-NCDE model, by focusing on the correlation between sampling points and their position information, not only further improves fitting accuracy under each sampling number but also mitigates the impact of sparse measurements on reconstruction results. The RMSE results indicate that PRN-NCDE enhances estimation accuracy by 54%, 58.5%, and 63.7% compared to RNN-NCDE when the sampling numbers are 60, 40, and 20, respectively. Moreover, when the sampling number decreases from 60 to 20, PRN-NCDE's prediction error increases by only 50%, remaining superior to the other two methods. Therefore, the PRN-NCDE model is more effective in handling irregular sequence reconstruction for controlled systems and can significantly reduce the impact of sparse measurements on reconstruction results.