An evolutionary approach for discovering non-Gaussian stochastic dynamical systems based on nonlocal Kramers-Moyal formulas

TL;DR

This paper introduces an evolutionary symbol sparse regression method utilizing nonlocal Kramers-Moyal formulas to identify non-Gaussian stochastic dynamical systems from data, effectively handling complex functional forms and Lévy noise.

Contribution

It develops a novel approach combining genetic programming, sparse regression, and nonlocal Kramers-Moyal formulas for discovering non-Gaussian stochastic systems from sample data.

Findings

Successfully applied to illustrative models

Demonstrates ability to uncover complex non-Gaussian dynamics

Offers a versatile tool for data-driven stochastic modeling

Abstract

Discovering explicit governing equations of stochastic dynamical systems with both (Gaussian) Brownian noise and (non-Gaussian) L\'evy noise from data is chanllenging due to possible intricate functional forms and the inherent complexity of L\'evy motion. This present research endeavors to develop an evolutionary symbol sparse regression (ESSR) approach to extract non-Gaussian stochastic dynamical systems from sample path data, based on nonlocal Kramers-Moyal formulas, genetic programming, and sparse regression. More specifically, the genetic programming is employed to generate a diverse array of candidate functions, the sparse regression technique aims at learning the coefficients associated with these candidates, and the nonlocal Kramers-Moyal formulas serve as the foundation for constructing the fitness measure in genetic programming and the loss function in sparse regression. The efficacy and capabilities of this approach are showcased through its application to several illustrative models. This approach stands out as a potent instrument for deciphering non-Gaussian stochastic dynamics from available datasets, indicating a wide range of applications across different fields.