## 5 Conclusions  
This paper takes the centring process of high-precision optical components as the research object, based on the proposed digital twin system for high-precision optical components and based on the five-dimensional model of the digital twin. This is in addition to, designing the four-layer architecture including the hardware layer, data layer, virtual layer, and application layer, respectively, and carrying out research and case development validation on the method of constructing high-fidelity digital twins, the mapping of the twin data, and the intelligent monitoring of critical components.  
The figure shows the results by comparing the VMD-GRU intelligent monitoring method proposed in this article with the prediction results of other methods. Regarding training time, the proposed method is 18 s, better than LSTM's 26 s and slightly worse than GRU's 15 s. In terms of measuring prediction accuracy, we chose RMSE and MAE as the two indicators, and it can be seen that the method proposed in this article achieved the best results of 10.6 and 7.95. However, considering the relatively small data volume (<1000) in this experiment, the performance of the LSTM method has not been fully demonstrated. Therefore, the method proposed in this article may not have the apparent superiority in prediction accuracy compared to LSTM, as shown in the figure.  
This study combines the processing of high-precision optical components from the perspective of the system construction of the proposed digital twin framework, again based on high-fidelity models, and the visualisation of the processing process and other services, not only for the application of digital twins in the field of high-precision machining to provide a demonstration, but also to practise Industry 4.0 in the manufacturing industry as for the in-depth application of information and communication technology to promote the concept of manufacturing intelligence; to accelerate the realisation of Industry 4.0, a preliminary exploration has been made. However, in this study, the model and accuracy and operational efficiency of the system needs to be further improved. At the same time, understanding how to optimise the construction process of the digital twin system, reduce its development cost, and use AR and other technologies to improve its interactive performance to minimise the difficulty of use, is a significant development direction for the future of the digital twin system of high-precision optical component machining.  
Figure 13: Evaluation results.  
**Author Contributions:** Conceptualization, Z.C.; methodology, Z.C. and Y.L.; validation, Z.C., Y.L. and Y.L.; writing--review and editing, Z.C., Y.L., G.M., Y.W. and B.Q.; supervision, Y.L.; project administration, Y.L.; funding acquisition, Y.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** The authors are grateful for the financial support from the National Key R&D Program of China (2022 YFB3706604) and Sichuan Provincial Science and Technology Achievement Transfer and Transformation Demonstration Project (2023 ZHCG0029); and Sichuan Science and Technology Program (2022 YFG0064).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data are available from the corresponding author on reasonable request. The data are not publicly available due to privatcy.  
**Conflicts of Interest:** The authors declare no conflict of interest.