Automating the Discovery of Partial Differential Equations in Dynamical Systems

In recent years, scientists have increasingly employed statistical and machine learning methods to uncover the governing equations of dynamical systems, particularly differential equations, from observational data [1-5]. Data-driven methods offer several advantages over traditional approaches that rely on first principles and expert knowledge. These methods can reveal patterns and relationships in the data that may not be apparent from first principles, providing new insights into complex systems [6, 7]. They are also adept at working with noisy or incomplete data commonly encountered in real-world applications, employing techniques from machine learning to enhance the robustness of discoveries [8-11]. Furthermore, by reducing the need for manual intervention and domain expertise, data-driven methods can significantly streamline the discovery process [12]. Data-driven discovery in dynamical systems has evolved from early parameter estimation using spline approximation and system reconstruction [13, 14], to leveraging statistical methods such as least squares [15-17], mixed-effects models [18, 19], and Bayesian approaches [2, 20] for parameter estimation in ordinary and partial differential equations (ODEs and PDEs). The advent of high-performance computing has further propelled symbolic regression, enabling the discovery of governing equations from data in physics and engineering [1, 21-23]. A notable development in this field is the Sparse Identification of Nonlinear Dynamics (SINDy) approach [3, 4], which constructs an extensive library of potential terms and employs the Sequential Threshold Ridge Regression (STRidge) algorithm [4] to select significant terms iteratively.