# Identification of High-Order Nonlinear Coupled Systems Using a Data-Driven Approach
## 1 Introduction  
### Motivation  
Understanding the differential equations that govern the dynamics of nonlinear systems in real-time is always considered advantageous in scientific research. However, the availability of accurate mathematical models for such systems is not always guaranteed. Despite the recognized benefits of having access to these equations, practical limitations and inherent complexities often prevent the simple formulation of mathematical representations for real-world nonlinear systems. Consequently, many authors have focused their efforts on the practical identification of mathematical models for nonlinear systems [1]. On the other hand, the continuously growing capabilities of computing and data acquisition equipment have produced an explosion of data-driven models, which in many cases are more appealing than those obtained by analytical modeling approaches [2]. Ahead of the data-driven modeling techniques is Deep Learning (DL) [3, 4]. DL has not only shown notable performance in image classification but is also very effective in predicting the future states of dynamical systems. However, its main drawback is the difficulty relating the DL models to the governing equations of the physical systems. A different data-driven approach is based on symbolic regression to obtain the structure of a nonlinear system from data [5]. Although this method identifies the interpretable physical models very well, its main drawback is the computational cost of working with a symbolic regressor, and it grows significantly as the dimension of the problem does. A genetic algorithm, considering the transition matrix in its fitness function, has been successfully applied to identify the physics-based longitudinal dynamics of an aircraft. However, itsmain drawback is that it is an iterative process strongly dependent on the size of the search region for parameters [6]. In [7], the problem has been restated as sparse identification combined with the Levenberg-Marquardt (LM) algorithm. [PERSON] first introduced the main idea for robust identification, which combines least squares minimization and discrimination through the \(L_{1}\) norm, and named it Shrinkage Lasso Selection [8]. However, the problem has been simplified even more by the Sparse Identification of Nonlinear Dynamics (SINDY), which combines sparse regression with the advantages of the verified efficacy of the least squares (LS) approach [9; 10]. [PERSON] et al. [9] proposed a method called Sparse Identification of Non-Linear Dynamics in 2016 which transforms a nonlinear model of the form \(\dot{x}=f(x,u)\), where \(x\in\mathbb{R}^{n}\) is the state vector and \(u\in\mathbb{R}^{p}\) is the input vector, into an equivalent linear model of the form \(\dot{X}=\Theta(X,U)\Xi\) (please refer to Section 2.2 for a more detailed explanation of this concept). This simplifies the nonlinear identification problem to solving a linear system with more equations than unknowns, which is a classical problem in linear algebra, namely, \(Ax=b\). This method offers an elegant approach to identifying nonlinear dynamics from data. The method can handle complex systems with perfect data from ideal experiments, as demonstrated in [9]. However, for simulations with imperfect data, the method requires some modifications, which are explained in [9]. The robustness analysis of the method with respect to the choice of \"hyperparameters\" is presented in [11]. The hyperparameters are related to the Lasso-type optimization, which is a classical optimization problem that uses the \(L_{1}\) norm instead of the \(L_{2}\) norm, as described in [12]. The extension of the method to systems with control inputs is discussed in [9]. SINDY has been applied during the identification of the models of complex systems as chaotic attractors, even in the presence of control signals, in a very successful way [9]. However, the original SINDY algorithm proves inadequate for high-order nonlinear systems with coupled dynamics, even when the structure of the mathematical model is known.