# A Novel Reconstruction Method for Irregularly Sampled Observation Sequences for Digital Twin
## 5 Conclusions  
The uncertainty and randomness in network communication and the non-synchronous sampling of sensors can cause irregularities, sparsity, and misalignment in measurement data. Existing methods struggle to achieve ideal reconstruction under sparse measurements and high noise, making them unsuitable for sequence reconstruction tasks in digital twin systems. To address these issues, this article establishes a variational autoencoder model based on parallel reference networks and neural controlled differential equations, which can effectively handle the reconstruction of noisy, irregular multivariate sequences in digital twin systems. Firstly, a multi-channel self-attention module is incorporated into the encoder. This module not only analyzes the position information of sampled data in the sequence and focuses on the correlation between data to enhance reconstruction accuracy under sparse measurements but also handles the misalignment and irregularity of observation sequences. Secondly, a parallel reference network is established. The reference sequence provided by the digital twin model is input and mapped to reference latent features, which are then fused with the latent features of the actual observation sequence in a weighted manner to address the reconstruction of observation sequences under high noise conditions. Thirdly, a decoder is constructed using an NCDE network, which takes into account the effect of control inputs on outputs to improve the sequence reconstruction performance of controlled systems. Finally, a weighted loss function for training the PRN- NCDE model is formed based on the calculated feature weights, which helps to better train the network parameters of the model. Simulation results show that the proposed PRN-NCDE model improves the estimation accuracy by more than 50% and 70%, respectively, compared with RNN-NCDE under different sampling numbers and noise levels, and by more than 80% and 60%, respectively, compared with RNN-NODE. It can recover the changing trend of observation sequence more accurately. Therefore, the proposed method can effectively improve the reconstruction accuracy of observation sequences under sparse measurements and large noise conditions, and is suitable for irregular sequence reconstruction tasks in digital twin systems with uncertain network transmission and non-synchronous sensor sampling. In future work, three aspects could be further investigated. First, the structure of the deep learning network can be improved to realize better abilities of feature extraction and prediction accuracy, especially under small sampling numbers and high noise levels. Second, the situation when outliers exist in the measurements should be considered to enhance the robustness of the reconstruction method. Third, we assume that the measurement noise follows Gaussian distribution. Non-Gaussian distribution should be considered for modelling measurement noise and the reconstruction performance under this situation could be another interesting research direction. Conceptualization, H.J. and Y.Z. ; methodology, H.J., Y.Z. and Y.C. ; software, H.J., Y.Z. and Q.Z. ; validation, H.J., Y.Z. and Q.Z. ; formal analysis, H.J. and Y.Z. ; investigation, H.J. and Y.Z. ; resources, H.J. and Y.Z. ; data curation, H.J. and Y.Z. ; writing--original draft preparation, H.J. and Y.Z. ; writing--review and editing, Q.Z. and Y.C. ; visualization, H.J. and Y.Z. ; supervision, Y.C. ; project administration, H.J. ; funding acquisition, H.J. All authors have read and agreed to the published version of the manuscript. This research was funded in part by the National Natural Science Foundation of China under Grant 62203349, and in part by the China Scholarship Council under Grant 201806280305. Not applicable. Data Availability Statement:Raw data were generated using the Matlab simulator. The data presented in this study are available on request from the corresponding author. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.