## 6 Conclusions  
This research developed a NARX network prediction model specifically tailored for assessing temperatures within seasonally frozen highway subgrades. Monitoring data such as temperature, water content, and frost heave collected from the subgrade of the Golmud-Naggu section on National Highway 109 from 2019 to 2020 were utilized as a case study. Both the NARX network and BP network models were employed for temperature prediction, and their results were compared with actual observed values to highlight the advantages of the NARX model over the traditional BP model. Additionally, temperatures at various subgrade depths were predicted to assess the model's accuracy and feasibility. Based on this comparative analysis, several key conclusions were drawn:  
1. Superior temperature predictive performance: The NARX network model demonstrated superior subgrade temperature predictive performance compared to the BP neural network. This is attributed to its strategy of implementing delays for the network input and output variables and preserving past values of time series variables. This approach provides the model with a robust memory capacity, enhancing the connection between time series data. Once the training of the open-loop structure was completed, the model was transitioned to a closed-loop structure for prediction. Given that the network under a closed-loop structure evolves into a dynamic recurrent neural network, its predictions were found to be more precise than those of the conventional BP neural network, with consistently stable performance. 2. Applicability at different depths: To further validate the applicability of the NARX network model, the prediction results of the two models at various subgrade depths were compared. The predictive performance of the NARX network model surpassed that of the BP network model across all subgrade depths, more closely mirroring actual temperature variations. These results suggest that the NARX network model possesses the capability to predict temperatures beneath seasonally frozen subgrades with enhanced accuracy, offering significant insights for practical engineering applications. 3. Implications and Future Work: This study offers valuable insights for the design and maintenance of subgrades in seasonally frozen regions by demonstrating the efficacy of the NARX network model in enhancing temperature prediction accuracy. These advancements enable engineers to make better-informed decisions, leading to improved subgrade performance. Future research should integrate additional environmental variables and apply the model to diverse geographic regions to further validate and enhance its predictive capabilities. Additionally, expanding the model's scope to include long-term predictions would be beneficial. Finally, collaborative efforts with field engineers and practitioners are essential to validate the model's practical utility and integrate it into routine subgrade maintenance and monitoring practices. Conceptualization, X.L. ; methodology, C.Z., X.L., and L.C. ; software, L.C. ; validation, X.L., L.C., X.H., and Z.K. ; formal analysis, C.Z. and Z.K. ; investigation, X.L., L.C., X.H., and Z.K. ; resources, C.Z. ; data curation, C.Z. and Z.K. ; writing--original draft preparation, C.Z., X.L., and L.C. ; writing--review and editing, C.Z. and X.L. ; visualization, L.C., X.L., and X.H. ; supervision, X.L. and X.H. ; project administration, C.Z. ; funding acquisition, X.L. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China (NSFC), grant number 42072314, and China Communications Construction Company Second Highway Consultant Co., Ltd. grant number KJFZ-2018-049. The APC was funded by China Communications Construction Company Second Highway Consultant Co., Ltd., grant number KJFZ-2018-049. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.