Governing equation discovery of a complex system from snapshots

TL;DR

This paper introduces SpIDES, a novel data-driven framework that discovers stochastic differential equations governing complex systems directly from snapshot data without requiring trajectory information, improving understanding and prediction of such systems.

Contribution

The paper presents a new simulation-free method, SpIDES, that combines probability flow reconstruction, density estimation, and Bayesian sparse identification to uncover SDEs from snapshot data, overcoming traditional assumptions.

Findings

Successfully identified the governing SDE of an over-damped Langevin system

Extracted interpretable drift and diffusion terms from data

Enhanced system understanding and predictive capabilities

Abstract

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.