Weak Collocation Regression for Inferring Stochastic Dynamics with Lévy Noise (Communications in Computational Physics, 2025)
With the rapid increase of observational, experimental and simulated datafor stochastic systems, tremendous efforts have been devoted to identifying governinglaws underlying the evolution of these systems. Despite the broad applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approachesto extracting stochastic dynamics with Lévy noise are relatively few. In this work, wepropose a Weak Collocation Regression (WCR) to explicitly reveal unknown stochasticdynamical systems, i.e., the Stochastic Differential Equation (SDE) with both $α$-stableLévy noise and Gaussian noise, from discrete aggregate data. This method utilizes theevolution equation of the probability distribution function, i.e., the Fokker-Planck (FP)equation. With the weak form of the FP equation, the WCR constructs a linear system of unknown parameters where all integrals are evaluated by Monte Carlo methodwith the observations. Then, the unknown parameters are obtained by a sparse linear regression. For a SDE with Lévy noise, the corresponding FP equation is a partialintegro-differential equation (PIDE), which contains nonlocal terms, and is difficult todeal with. The weak form can avoid complicated multiple integrals. Our approach cansimultaneously distinguish mixed noise types, even in multi-dimensional problems.Numerical experiments demonstrate that our method is accurate and computationallyefficient.
Keywords: Weak collocation regression; learning stochastic dynamics; Lévy process; Fokker-Planck equations; weak SINDy.