Kernel-based learning framework for discovering the governing equations of stochastic jump-diffusion processes directly from data

Wenqing Sun, Jinqian Feng, Jin Su, and Qin Guo

Phys. Rev. E 108, 035306 – Published 15 September 2023

Abstract

Discovering the underlying mathematical-physical equations of complex systems directly from observational data has been a challenging inversion problem. We propose a data-driven framework for identifying dynamical information in stochastic diffusion or stochastic jump-diffusion systems. The probability density function is utilized to relate the Kramers-Moyal expansion to the governing equations, and the kernel density estimation method, improved by the Fourier transform idea, is used to extract the Kramers-Moyal coefficients from the time series of the state variables of the system. These coefficients provide the data expression of the governing equations of the system. Then a data-driven sparse identification algorithm is used to reconstruct the underlying dynamic equations. The proposed framework does not rely on prior assumptions, and all results are obtained directly from the data. In addition, we demonstrate its validity and accuracy using illustrative one- and two-dimensional examples.