The paper "Discovering governing equations from data: Sparse identification of nonlinear dynamical systems" by Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz introduces a method to discover governing physical equations from measurement data using sparse regression and machine learning techniques. The authors assume that the governing equations are sparse in the space of possible functions, meaning that only a few terms are important for describing the dynamics. This assumption is valid for many physical systems. The method uses sparse regression to determine the fewest terms required to accurately represent the data, resulting in parsimonious models that balance complexity with descriptive ability while avoiding overfitting. The paper demonstrates the algorithm on various problems, including simple canonical systems (linear and nonlinear oscillators, the Lorenz system) and more complex systems like fluid vortex shedding behind an obstacle. The fluid example illustrates the method's ability to uncover the underlying dynamics of a system that took experts nearly 30 years to resolve. The method is also shown to generalize to parameterized, time-varying, or externally forced systems. The authors discuss the background of system identification techniques and the challenges of extracting physical laws from data. They highlight the limitations of traditional methods and introduce sparse regression as a solution. The method is compared to symbolic regression, which is more expensive and less scalable. The paper also covers extensions to discrete-time systems, high-dimensional systems, and systems with external forcing or bifurcation parameters. The results section presents several examples, including a two-dimensional damped oscillator, a three-dimensional linear system, the Lorenz system, and fluid wake behind a cylinder. The method accurately identifies the governing equations and captures the attractor dynamics, even in the presence of noise. The paper concludes by discussing the potential applications of the method in various fields and open problems related to dynamical systems.The paper "Discovering governing equations from data: Sparse identification of nonlinear dynamical systems" by Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz introduces a method to discover governing physical equations from measurement data using sparse regression and machine learning techniques. The authors assume that the governing equations are sparse in the space of possible functions, meaning that only a few terms are important for describing the dynamics. This assumption is valid for many physical systems. The method uses sparse regression to determine the fewest terms required to accurately represent the data, resulting in parsimonious models that balance complexity with descriptive ability while avoiding overfitting. The paper demonstrates the algorithm on various problems, including simple canonical systems (linear and nonlinear oscillators, the Lorenz system) and more complex systems like fluid vortex shedding behind an obstacle. The fluid example illustrates the method's ability to uncover the underlying dynamics of a system that took experts nearly 30 years to resolve. The method is also shown to generalize to parameterized, time-varying, or externally forced systems. The authors discuss the background of system identification techniques and the challenges of extracting physical laws from data. They highlight the limitations of traditional methods and introduce sparse regression as a solution. The method is compared to symbolic regression, which is more expensive and less scalable. The paper also covers extensions to discrete-time systems, high-dimensional systems, and systems with external forcing or bifurcation parameters. The results section presents several examples, including a two-dimensional damped oscillator, a three-dimensional linear system, the Lorenz system, and fluid wake behind a cylinder. The method accurately identifies the governing equations and captures the attractor dynamics, even in the presence of noise. The paper concludes by discussing the potential applications of the method in various fields and open problems related to dynamical systems.