Governing equation discovery of a complex system from snapshots

Complex systems in physics, chemistry, and biology that evolve over time with inherent randomness are typically described by stochastic differential equations (SDEs). A fundamental challenge in science and engineering is to determine the governing equations of a complex system from snapshot data. Traditional equation discovery methods often rely on stringent assumptions, such as the availability of the trajectory information or time-series data, and the presumption that the underlying system is deterministic. In this work, we introduce a data-driven, simulation-free framework, called Sparse Identification of Differential Equations from Snapshots (SpIDES), that discovers the governing equations of a complex system from snapshots by utilizing the advanced machine learning techniques to perform three essential steps: probability flow reconstruction, probability density estimation, and Bayesian sparse identification. We validate the effectiveness and robustness of SpIDES by successfully identifying the governing equation of an over-damped Langevin system confined within two potential wells. By extracting interpretable drift and diffusion terms from the SDEs, our framework provides deeper insights into system dynamics, enhances predictive accuracy, and facilitates more effective strategies for managing and simulating stochastic systems.