Automated Discovery of Continuous Dynamics from Videos

Dynamical systems are predominantly described by physical laws, which involve a set of physical variables to represent the system's states and a set of equations to connect these variables to model the system's evolution over time. To uncover these physical laws from natural phenomena, scientists begin by identifying the appropriate physical variables, such as position and velocity in classical mechanics, electric current and magnetic induction in electromagnetism, and pressure and velocity fields in fluid mechanics, and measure them from raw observations of the system. Subsequently, they derive equations from these variables to articulate the underlying physical principles, such as Newton's laws in classical mechanics, Maxwell's equations in electromagnetism, and the Navier-Stokes equations in fluid mechanics. By applying various mathematical tools to analyze these equations, scientists gain a profound understanding of natural phenomena and can predict the system's future behaviors. This paradigm of scientific discovery, most well-known since the work of Tycho Brahe and Johannes Kepler from more than 400 years ago, has been remarkably successful across almost all areas of modern science. Despite centuries' efforts, using a similar paradigm to discover physical variables and equations for new systems still remains challenging, as seen in early attempts to automate the process of discovering equations from given physical variables [1-3].