  With the rapid increase of observational, experimental and simulated data for
stochastic systems, tremendous efforts have been devoted to identifying
governing laws underlying the evolution of these systems. Despite the broad
applications of non-Gaussian fluctuations in numerous physical phenomena, the
data-driven approaches to extracting stochastic dynamics with L\'{e}vy noise
are relatively few. In this work, we propose a Weak Collocation Regression
(WCR) to explicitly reveal unknown stochastic dynamical systems, i.e., the
Stochastic Differential Equation (SDE) with both $\alpha$-stable L\'{e}vy noise
and Gaussian noise, from discrete aggregate data. This method utilizes the
evolution 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 method with the observations. Then, the unknown
parameters are obtained by a sparse linear regression. For a SDE with L\'{e}vy
noise, the corresponding FP equation is a partial integro-differential equation
(PIDE), which contains nonlocal terms, and is difficult to deal with. The weak
form can avoid complicated multiple integrals. Our approach can simultaneously
distinguish mixed noise types, even in multi-dimensional problems. Numerical
experiments demonstrate that our method is accurate and computationally
efficient.
