## 1 Introduction  
Many existing bridges in urban rail transit were designed according to outdated regulations and guidelines, which may not fully accommodate current and emerging train configurations [1]. With rapid urbanization and the expansion of rail transit networks, both the axle load and speed of urban rail trains are increasing significantly. For daily maintenance of urban rail transit bridges, it is essential to understand the axle loads of moving trains, especially their extreme values [2, 3]. Therefore, it is critical to accurately identify train axle loads and evaluate their effect on bridge structures. This not only facilitates the updating of current design specifications but also enhances safe operation and extends the service life of existing bridges in urban rail transit networks. In general, there are two distinct methods to identify train axle loads running on rail transit bridges, namely the direct method and the indirect method. The direct method utilizes advanced sensors to directly measure the contact force between the wheel and rail of moving trains, while the indirect method infers train axle loads from measured structural responses of rails or bridges. Thus, the direct approach hugely depends on advanced sensors installed on the components of railway trains, particularly on wheels and bogies or on the bottom rails. For instance, based on electrical strain gauges installed on wheelsets, [PERSON] and [PERSON] [4] developed a method to measure train wheel load at the rail-wheel contact point. Similarly, [PERSON] et al. [5] introduced a method to detect wheel-rail contact forces without the need for special wheelsets equipped with strain gauges, slip rings, or telemeters. Additionally, [PERSON] et al. [6] conducted field measurements of the vertical wheel-rail contact forces within the frequency range of 0-2000 Hz. Another study by [PERSON] et al. [7] utilized instrumented wheels with strain gauges glued to them, incorporating a novel signal-processing method to suppress disruptive wheel resonances. Furthermore, a system based on dynamometric wheelsets equipped with strain gauges and distance lasers was proposed to measure wheel-rail contact forces. [PERSON] [8] extended this approach by placing strain sensors not only on wheels but also on train axles to evaluate wheel-rail contact forces and axle stresses in high-speed trains. However, these advancements are accompanied by significant challenges, such as the inability to monitor axle loads across many trains and stringent maintenance requirements for sensors installed on train components. Hence, several researchers have addressed the challenge of measuring train axle loads by deploying various sensor technologies on the bottom rails. [PERSON] et al. [9] employed a sophisticated strain gauge setup on the bottom rails, proposing a novel wayside approach to measuring and evaluating wheel-rail contact forces and positions. To obtain a more reliable wheel-rail contact force, [PERSON] et al. [10] explored a continuous measurement method using shear techniques, validating their approach with an indoor prototype. Due to the development of detection technology, some advanced sensors have also been installed on bottom rails to detect the wheel-rail contact force. In their study [11], a fiber Bragg grating (FBG) sensor array system, which was installed on rails over a long range, was presented for the real-time monitoring of railway train axle loads. [PERSON] et al. [12] utilized Distributed Acoustic Sensing, an optical sensing technology, to measure rail strains accurately, reconstructing continuous vertical wheel-rail forces. For all the aforementioned studies using the direct method, sensors mounted on train components or bottom rails are susceptible to damage from the wheel impact of high-speed trains during operation. Consequently, the direct measurement method faces challenges in meeting the long-term monitoring needs of train axle loads for urban rail transit bridges.