# A Novel Reconstruction Method for Irregularly Sampled Observation Sequences for Digital Twin
## 1 Introduction
In 2018, [PERSON] et al. introduced the NODE network [22], offering new solutions for irregular sequence reconstruction. Subsequently, [PERSON] et al. [23] developed a latent NODE model, which combines RNN and NODE in a variational autoencoder (VAE) framework, outperforming the exponential decay RNNs. The GRU-ODE-Bayes model proposed by [PERSON] et al. [24] achieved good results on sparse data. [PERSON] et al. [25] presented LG-ODE, which is a latent ordinary differential equation VAE for multi-agent systems with known graph structure. LG-ODE can learn the embedding of high-dimensional trajectories and deduce the latent system dynamics simultaneously. In practical applications, most systems have state trajectories influenced not only by initial values but also by control inputs. NODEs cannot account for control inputs' impact on output sequences, limiting their application in controlled system sequence reconstruction. Moreover, when the measurement data are sparse or noisy due to network issues and electromagnetic interference, existing methods struggle to handle digital twin sequence reconstruction tasks. To address these challenges, this article proposes a VAE model based on a parallel reference network (PRN) and neural controlled differential equation (NCDE) [26] for noisy multivariate irregular sequence reconstruction in controlled systems. First, we establish a multi-channel self-attention (MCSA) module to analyze the position information and correlations of sampled data in the target sequence, improving reconstruction accuracy under sparse measurements while effectively handling misalignment and irregularity through multi-channel and masking mechanisms. Second, to enhance reconstruction accuracy under high noise levels, we construct a PRN to obtain reference features from the prediction results of digital twin model and fuse them with actual data features. We also calculate feature weights based on the noise level of the observation sequence to determine the proportion of actual and reference information in the fused features. Third, we use NCDE to build a decoder that can predict observations at any time by incorporating control inputs to solve the sequence reconstruction problem in controlled systems. Finally, we develop a weighted loss function based on feature weights to better train the model's network parameters. Simulation experiments demonstrate the effectiveness and fitting accuracy of the proposed model for controlled systems under sparse measurements and high noise levels compared to the existing methods. Overall, the main contributions of this article are threefold:  
1. We propose a PRN-NCDE-based VAE model that improves sequence reconstruction accuracy for controlled systems under sparse measurements and high noise levels. 2. We develop an MCSA module that can not only analyze data position and correlations to enhance reconstruction performance under sparse measurements, but also effectively handle misaligned and irregular observation sequences. 3. To improve reconstruction accuracy under high noise levels, we establish a PRN to obtain reference features and calculate feature weights based on the noise level of observation data for weighted fusion of latent features. The rest of this article is organized as follows. Section 2 formally describes the irregular sequence reconstruction problem. Section 3 presents the overall framework of the PRN-NCDE model and analyzes the MCSA module, PRN, NCDE network, and weighted loss function. Section 4 validates effectiveness of the proposed model compared to the existing methods through a boiler system under different sampling numbers and high noise levels. Conclusions and future work are discussed in Section 5.