Extracting Non-Gaussian Governing Laws from Data on Mean Exit Time

TL;DR

This paper presents a novel method to extract non-Gaussian governing laws from data using mean exit time observations, applicable to complex stochastic systems driven by Levy motion, through sparse regression and inverse problem solving.

Contribution

It introduces a new approach combining sparse regression and inverse problem techniques to identify stochastic differential equations from mean exit time data, including non-Gaussian Levy-driven systems.

Findings

Method accurately recovers governing laws from simulated data.

Applicable to both Gaussian and non-Gaussian Levy-driven systems.

Works with systems having complex rational drift terms.

Abstract

Motivated by the existing difficulties in establishing mathematical models and in observing the system state time series for some complex systems, especially for those driven by non-Gaussian Levy motion, we devise a method for extracting non-Gaussian governing laws with observations only on mean exit time. It is feasible to observe mean exit time for certain complex systems. With the observations, a sparse regression technique in the least squares sense is utilized to obtain the approximated function expression of mean exit time. Then, we learn the generator and further identify the stochastic differential equations through solving an inverse problem for a nonlocal partial differential equation and minimizing an error objective function. Finally, we verify the efficacy of the proposed method by three examples with the aid of the simulated data from the original systems. Results show that the method can apply to not only the stochastic dynamical systems driven by Gaussian Brownian motion but also those driven by non-Gaussian Levy motion, including those systems with complex rational drift.