## 1 Introduction  
Mathematical model building, especially the construction of governing equations, is a significant problem for a large variety of dynamic systems, such as vibration systems [1], biological systems [2], and fluid mechanical systems [3]. These models can be used for prediction, simulation, condition monitoring, optimization, etc. [4]. The traditional methods to build a model are based on the basic knowledge in one or many categories. These types of model building are described as \"bottom-up\" [5]. However, these methods work only with _a priori_ knowledge. In recent years, the rapid growth of computational capacity of computers has enabled researchers in several domains to build models with a \"top-down\" approach using acquired data instead of the domain knowledge [5]. Many topics are related to this approach, such as System Identification (SI) and the development or improvement of structural models using the input and output data from real measurements [6]. These methods are especially useful when the first principle knowledge is not discovered, for example, the macroscopic behavior of advanced materials [7], in aerospace engineering [8], in biology [2, 9, 10], in economy [11], in music [12, 13], etc. The models of vibration systems can be identified with various approaches, such as iterative model updating [14], the Polynomial NonLinear State Space (PNLSS) model [15], Volterra series [16], the Wiener and Hammerstein model [17, 18], artificial neural networks [19, 20], and equation-free model building [21]. On the downside, these methods usually build \"black-box\" or \"grey-box\"models, which means the theoretical understanding of the system is totally or partially absent from the model [4]. In the category of sparse regression, the SINDy framework (Sparse Identification of Nonlinear Dynamical Systems) shows a bright perspective in terms of interpretability. It is a technique to build the differential governing equation by extracting a few relevant terms from a library of candidate functions [22]. The polynomial development of a time series \(\mathbf{x}(t)\) is assumed as candidate functions and the regression method _Least Squares method with Sequential Threshold_ (LSST, also called STLS [23]) is suggested for sparse regression [24]. In recent research, SINDy was applied to boundary value problems [25], and advanced materials [7]. Furthermore, SINDy is also applicable to Model Predictive Control (MPC) to reduce complexity and to enhance the interpretability of the integrated empirical model [26; 27]. Many extensions have been adopted on the SINDy framework for different benefits. First, this framework can be modified using an enlarged library of candidate functions, such as trigonometric functions [24], partial differential equations [28; 29], and rational functions [22]. Second, different methods of sparse regression can be applied. The methods, for example, _Least Absolute Shrinkage and Selection Operator_ (LASSO) [30] and compressed sensing [31; 32; 33], have been used to enforce sparsity before the development of SINDy. Recently, a broad range of regularized regression methods have been reformulated and generalized in the framework of Sparse Relaxed Regularized Regression (SR3) [34]. Throughout this work, LSST and the _Least Squares method Post-LASSO_ (LSPL) [35] will be discussed. Third, different signal processing methods are used to correct or transfer the measurement in order to make the solved problem better-conditioned, such as normalization [29], Hankel Alternative View Of Koopman (HAVOK) [36], and neural networks [37]. Since the governing equation of a discrete vibration system and SINDy share the same mathematical model structure, the SINDy framework gains attention for identifying the governing equations of discrete vibration systems.