Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas

Abstract

With the rapid development of computational techniques and scientific tools, great progress of data-driven analysis has been made to extract governing laws of dynamical systems from data. Despite the wide occurrences of non-Gaussian fluctuations, the effective data-driven methods to identify stochastic differential equations with non-Gaussian Lévy noise are relatively few so far. In this work, we propose a data-driven approach to extract stochastic governing laws with both (Gaussian) Brownian motion and (non-Gaussian) Lévy motion, from short bursts of simulation data. Specifically, we use the normalizing flows technology to estimate the transition probability density function (solution of nonlocal Fokker-Planck equation) from data, and then substitute it into the recently proposed nonlocal Kramers-Moyal formulas to approximate Lévy jump measure, drift coefficient and diffusion coefficient. We demonstrate that this approach can learn the stochastic differential equation with Lévy motion. We present examples with one- and two-dimensional, decoupled and coupled systems to illustrate our method. This approach will become an effective tool for discovering stochastic governing laws and understanding complex dynamical behaviors.

References in corpus (21)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Data-driven discovery of coordinates and governing equations
• Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system
• Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
• Data-driven model reduction and transfer operator approximation
• Time-series learning of latent-space dynamics for reduced-order model closure
• Subspace dynamic mode decomposition for stochastic Koopman analysis
• Understanding the geometry of transport: diffusion maps for Lagrangian trajectory data unravel coherent sets
• Galerkin Approximation of Dynamical Quantities using Trajectory Data
• Coarse-Grained Kinetic Computations for Rare Events: Application to Micelle Formation
• Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach
• A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise
• Coarse-scale PDEs from fine-scale observations via machine learning
• An early warning indicator for atmospheric blocking events using transfer operators
• Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes
• The tipping times in an Arctic sea ice system under influence of extreme events
• Detecting the maximum likelihood transition path from data of stochastic dynamic systems
• Discovering transition phenomena from data of stochastic dynamical systems with Lévy noise
• Feller Processes: The Next Generation in Modeling. Brownian Motion, Lévy Processes and Beyond
• Maximal Likely Phase Lines for a Reduced Ice Growth Model

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