## 7 Conclusions  
Leveraging high-reliability and high-precision DT technology, this study proposes a virtual-real hybrid framework for bearing thermal prediction under complex operating conditions and long-term operation. The framework employs an NOA-based identification method to determine key thermal parameters in the LPTN model within the VML, addressing the time-varying and ambiguous characteristics of unknown parameters. This enables thermal prediction transitions from low-risk behaviors (low-speed, light-load) to high-risk behaviors (high-speed, heavy-load) in bearing systems. A comprehensive analysis of comparative results yields the following conclusions:  
1. The NOA-based identification mechanism achieves near-perfect alignment between identified and target values for time-varying and ambiguous parameters, with convergence speeds 10 times faster than the PSO-based.
2. Under variable-speed conditions, the proposed framework exhibits maximum temperature errors of 1.2 \({}^{\circ}\)C and 0.5 \({}^{\circ}\)C at reference nodes during identification. With 270 samples, the average Euclidean distance between twin and validation temperatures reaches a minimum of 4.9 K. Expanding the number of reference nodes from 4 to 13 reduces the average Euclidean distance from 13.7 K to 0.4 K, validating its capability for thermal prediction from low-speed to high-speed and highlighting the benefits of multi-speed-range samples. The uncertainty analysis of the rapid acceleration test shows that all real temperature data points fall within the 95% prediction intervals of the predicted temperatures, confirming the robust predictive capability of the implemented framework.
3. Under variable radial load conditions, steady-state temperature errors at reference nodes are 0.6 \({}^{\circ}\)C and 0.4 \({}^{\circ}\)C for 2650 N light-load identification. When predicting the temperature of heavy-load conditions (4000 N and 5500 N), maximum differences remain below 2.6 \({}^{\circ}\)C, and stabilization stage errors \(\leq\) 3.5% even at 5500 N. But uncertainty analysis shows that real temperature points intermittently fall outside the 95% prediction intervals; the proposed prediction framework requires further enhancement in accuracy under variable load conditions.  
Conceptualization, J.C. ([PERSON]); methodology, Y.W.; software, Y.W.; validation, J.C. ([PERSON]), Q.T.; formal analysis, L.Y.; investigation, L.Y.; resources, Q.T.; data curation, J.C. ([PERSON]); writing--original draft preparation, Y.W.; writing--review and editing, Y.W.; visualization, Q.T.; supervision, J.C. ([PERSON]); project administration, L.Y.; funding acquisition, J.C. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key Research and Development Program of China, grant number 2022 YFB2502600, and the Science and Technology Research and Development Joint Fund of Henan Province, grant number 225101610002.  
Not applicable.  
The data are not publicly available due to confidentiality requirements of the project.  
Author [PERSON] was employed by the company Luoyang Bearing Research Institute Co., Ltd. Author [PERSON] was employed by the company CRRC Qingdao Sifang Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} DT & Digital Twin \\ NOA & Nutcracker Optimization Algorithm \\ LPTN & Lumped parameter thermal network \\ FE & Finite Element \\ CFD & Computational Fluid Dynamics \\ UDFs & User-defined functions \\ TRs & Thermal resistances \\ PEL & Physical Entity Layer \\ VML & Virtual Model Layer \\ DAL & Data Acquisition Layer \\ DMP & Data Mapping Part \\ CI & Confidence Intervals \\ PI & Prediction Intervals \\ \end{tabular}