## 1 Introduction  
Bridge load identification is a critical task in structural health monitoring (SHM), ensuring the safety and longevity of bridge infrastructures [1, 2], since many approaches of SHM in practice first need the bridge load information. The accurate determination of moving loads, such as vehicles or trucks, plays a pivotal role in assessing structural performance and detecting potential damage [3, 4]. Traditional methods for load identification often rely on dynamic response analysis, weigh-in-motion (WIM) systems, or data-driven approaches [5, 6]. However, these techniques may face challenges in cases involving unknown moving loads, complex structural behavior, or noisy measurement data. In recent years, the influence line (IL) concept has gained significant attention as an effective tool for load identification and damage detection in bridge structures. The influence line represents the structural response at a specific point due to a unit load moving across the bridge, providing valuable insights into load distribution and structural behaviors. [PERSON] et al. [7] proposed a probabilistic frequency-domain approach for IL extraction under unknown moving loads, demonstrating its potential for accurate load estimation. Similarly, [PERSON] et al. [8] and [PERSON], et al. [9] explored advanced machine learning techniques to enhance moving load identification, highlighting the growing role of artificial intelligence in this field. [PERSON] and [PERSON] [10] addressed this limitation by developing a method for the online tracking of the influence lines of high-speed railway bridges, utilizing only the dynamic response of passing trains. Unlike traditional approaches, which determine IL through static loading, [PERSON] et al. [11] proposed a novel technique to identify the rotating influence line (RIL) based on vehicle-bridge interaction (VBI) under stochastic traffic loading conditions. It is based on the assumption that the stiffness has no change in the work life of the bridge and there is no stiffness loss as time goes by. Several studies have utilized IL-based methods for axle load identification and bridge damage assessment. [PERSON] et al. [12] developed a method for identifying vehicle axle loads using bending moment influence lines, while [PERSON] et al. [13] implemented a fiber optic-based WIM system for train load identification. [PERSON] et al. [14] and [PERSON] et al. [15] further investigated numerical and experimental approaches for moving load identification, emphasizing the importance of dynamic response analysis. Additionally, [PERSON] and [PERSON] [16] and [PERSON] et al. [17] demonstrated the effectiveness of IL-based techniques in damage detection, leveraging both static and dynamic structural responses. [PERSON] et al. [18] developed a quasi-static method for identifying structural damage based on a sensor's line of influence (IL) and an empirical Bayesian threshold estimator and validated the feasibility of the method through numerical models and laboratory tests. [PERSON] et al. [19] proposed a new method of beam bridge damage identification based on multi-source influence line information fusion, aiming at the problem of poor robustness and accuracy of identification results caused by single measurement point data using the influence line class method, and conducted simulation verification with an actual steel-mixed composite beam bridge. Recent advancements have extended the application of influence lines to distributed sensing and long-gauge strain measurements. [PERSON] et al. [20,21] proposed damage identification methods based on strain envelope lines under moving loads, while [PERSON] et al. [22] and [PERSON] et al. [23] explored dynamic response processing and strain influence lines for bridge damage assessment. [PERSON] et al. [24] further introduced an equivalent thrust-influenced line method for tied-arch bridges, showcasing the versatility of IL-based approaches.