## 5 Conclusions  
The objective of this study was to reconstruct high-frequency bridge vibrations across different bridge cross-sections. The theoretical responses of a bridge in a simple VBI system were initially analyzed. The relationship between sections was then explicitly formulated to discuss its time-invariant characteristics under varying temperature and traffic conditions. Subsequently, a BP-ANN network was chosen for reconstructing the high-frequency bridge responses, leveraging these derived characteristics. A series of finite element models were developed to verify these characteristics. A field experiment was conducted on a continuous bridge to validate the proposed method. The main conclusions are summarized below as the following:  
1. Essentially, this relationship can be characterized as a transfer matrix that is associated with mode shapes. The target relationship should remain constant and time-invariant under varying traffic and temperature conditions.
2. The BP-ANN method proved to be an effective tool for reconstructing responses across various cross-sections. In the field test, the reconstruction result matched well with the monitoring data.
3. The proposed method facilitated the utilization of relatively short-term monitoring data to reconstruct long-term bridge responses effectively.  
In comparison to the preceding study, the proposed method is data-driven and does not necessitate the construction of finite element models, thereby enhancing its time efficiency for engineering applications. Moreover, this method permits long-term monitoring with a short-term monitoring database, thus obviating the necessity for a vast database and making it more accessible for implementation in real-world engineering scenarios. Furthermore, grounded in the bridge's physical characteristics, it is expected to be applicable to general short- and medium-span bridges.  
It is important to note that the current study only considered bending modes, and not torsional modes, which may also contribute to the bridge dynamics in bridges with multiple lanes. As a result, the current method might not encompass all critical vibrational modes. It is suggested that sensor placement be optimized accordingly. Moreover, this study was limited to bridges with constant mode shapes. However, the method may not be suitable for simulating relationships in some large-span bridges that exhibit non-linear time-dependent properties or have non-linear material components whose mode shapes may change over time. These concerns will be addressed in future studies.  
Conceptualization, methodology, software, validation, and writing--original draft preparation, X.L.; investigation, resources, and data curation, Y.X.; writing--review and editing, supervision, and funding acquisition, L.S. All authors have read and agreed to the published version of the manuscript.  
Figure 14: Scatter plot of high-frequency strain at opposite sides in cross-section 02.  
**Funding:** The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Shanghai Qi Zhi Institute [Grant No. SQZ202310].  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author due to [PERSON].  
**Conflicts of Interest:** The authors declare no conflicts of interest.