# A Novel Reconstruction Method for Irregularly Sampled Observation Sequences for Digital Twin
## 1 Introduction  
As one of the key technologies for the digital transformation of traditional industries, digital twin can create virtual mappings of physical entities, which enables real-time reflection of the entire lifecycle of physical systems and facilitates their optimization andmonitoring [1]. Maintaining consistency between the twin model and the physical system is crucial for the successful implementation of digital twins' functionalities [2]. However, several factors can degrade the training and prediction performance of twin models. First, network communication often has irregularities such as delays and packet loss, leading to irregular sampling intervals [3]. Second, sensors in physical systems have varying sampling periods, causing misalignment and sparsity in multivariate sequences. These issues make the sampled sequences unsuitable for model training and produce negative impacts on prediction performance [4]. Sequence reconstruction is an effective means for these problems in digital twin systems. It involves rebuilding irregular observation sequences using function fitting or generative neural networks to obtain regular sequence data. Traditional methods include polynomial fitting [5] and maximum likelihood estimation [6]. Polynomial fitting approximates complex functions by learning suitable polynomial coefficients but requires predefining the polynomial degree [7;8]. A higher predefined polynomial degree may lead to overfitting and a lower one may have negative impact on the fitting performance. For complex and ever-changing observation sequences, it is often difficult to make an assumption on its degree. Moreover, polynomial fitting is very sensitive to noise, resulting in its inapplicability for noise-corrupted observation sequences [9]. Based on the curve fitting method in statistics, maximum likelihood estimation determines model parameters by calculating likelihood function values but shares similar limitations with polynomial fitting, such as narrow applicability and the need to assume a model form. Narrow applicability means this method can only produce satisfactory results under some certain distributions [10]. And an assumed model form will prevent the method from dealing well with complex sequences influenced by control inputs. Fixed-time discretization is another conventional approach that divides continuous-time observations into fixed-length, non-overlapping data windows [11]. Although this method is easy to follow and utilize, it may induce empty windows and data aggregation issues when window sizes are increased [12]. Therefore, window size is a key parameter for this method. However, determining an appropriate window size for complex sequences is often difficult. Alternative solutions involve developing neural network models that can directly use irregular sequences as inputs, including interpolation-based and neural ordinary differential equation (NODE)-based methods. Interpolation-based methods use past and future observations for interpolation. [PERSON] et al. [13] studied a multi-directional recurrent neural network (RNN) that can use past and future observation data flows at a given time step to realize interpolation, thus improving estimation performance in missing measurement scenarios. [PERSON] et al. [14] developed an interpolation prediction network, which consists of several semi-parametric radial basis function (RBF) interpolation layers. This interpolation network can interpolate multivariate and irregularly time series against a set of reference time points. In [15; 16; 17], attention mechanism is used for time encoding and several multi-time attention-based RNNs are proposed for irregular sequence modeling. However, these interpolation-based methods face variable uncertainty when observation intervals change significantly [18]. Exponential decayed RNN autoencoder frameworks for sequence reconstruction were studied in [19; 20], but [PERSON] et al. [21] found these networks have no obvious improvements in prediction accuracy compared to the standard RNN autoencoders.