## 1 Introduction  
With the accumulation of service time, structures will be damaged to a certain extent, resulting in potential safety hazards. If structural damages can be found as early as possible during service, the service function and remaining life of the structure can be effectively improved. The existence of structural damage will generally cause changes in structural dynamic characteristics such as structural stiffness, frequency, or vibration mode. Structural damage identification is one of the inverse problems of structural health monitoring (SHM) [1]. [PERSON] et al. [2,3] briefly reviewed the field of SHM and discussed the future development trend and proposed a nested attribute-based few-shot meta-learning paradigm for structural damage identification. Structural damage identification has developed rapidly in recent decades. [PERSON] et al. [4] proposed the use of the instantaneous damping coefficient as a damage detection index combined with a wavelet transform; the method was verified in numerical simulations and laboratory tests. [PERSON] et al. [5, 6, 7] put forward some new methods for structural damage detection by developing a novel approach based on densely connected convolutional networks, using the higher-order response and phase space topology technology based on singular spectrum analysis, and improving the artificial bee colony algorithm. [PERSON] et al. [8] deduced a formula for calculating the first-order sensitivity of element modal strain energy and successfully applied to detect damage of numerical models. [PERSON] et al. [9, 10, 11, 12] put forward a series of new damage detection methods, respectively, using a multi-layer genetic algorithm, developing a geometric modal strain energy-eigenvalue ratio, and a double criteria method based on vibration characteristics to detect and locate the damage of various kinds of structures. [PERSON] and [PERSON] [13; 14; 15] reviewed the state-of-the-art approaches for the framework of vibration-based damage identification and proposed new methods for damage identification by using vehicle-bridge interaction analysis. [PERSON] et al. [16] introduced a damage diagnosis algorithm by using a sequential change-point detection method. [PERSON] and [PERSON] [17] compared several structural damage identification methods based on modal curvature, natural frequency, and modal strain and pointed out the identification ability of each method on the location and degree of damage. [PERSON] and [PERSON] [18] utilized the sensitivity of the frequency response function to effectively identify the location and degree of damage. [PERSON] et al. [19] proposed a multi-mode multi-directional damage index method for multi-directional damage detection in frame structures. [PERSON] et al. [20] proposed a new iterative regularization method and an improved sensitivity function for the identification of structural damage. [PERSON] et al. [21; 22] proposed two new methods to detect damage, respectively, based on the Bayesian probabilistic approach and a time series analysis with sparse regularization. [PERSON] et al. [23] proved that the square of the instantaneous amplitude is a good damage detection index for a simply supported beam on an elastic foundation. [PERSON] et al. [24] presented a damage detection method for masonry arches, and the frequency variations are used in this new method. [PERSON] et al. [25] presented a multi-scale reconstructed attractors-based method for the identification of structural damage. [PERSON] et al. [26] developed a novel structural damage detection method using factor analysis in the sparse Bayesian learning framework. [PERSON] et al. [27] proposed a novel damage identification approach for longitudinally vibrating structures. [PERSON] et al. [28] developed a damage identification approach based on strain mode differences using the inverse finite element method. [PERSON] et al. [29] used kernelized Bayesian transfer learning with the population-based SHM.