  With the rapid increase of valuable observational, experimental and
simulating data for complex systems, great efforts are being devoted to
discovering governing laws underlying the evolution of these systems. However,
the existing techniques are limited to extract governing laws from data as
either deterministic differential equations or stochastic differential
equations with Gaussian noise. In the present work, we develop a new
data-driven approach to extract stochastic dynamical systems with non-Gaussian
symmetric L\'evy noise, as well as Gaussian noise. First, we establish a
feasible theoretical framework, by expressing the drift coefficient, diffusion
coefficient and jump measure (i.e., anomalous diffusion) for the underlying
stochastic dynamical system in terms of sample paths data. We then design a
numerical algorithm to compute the drift, diffusion coefficient and jump
measure, and thus extract a governing stochastic differential equation with
Gaussian and non-Gaussian noise. Finally, we demonstrate the efficacy and
accuracy of our approach by applying to several prototypical one-, two- and
three-dimensional systems. This new approach will become a tool in discovering
governing dynamical laws from noisy data sets, from observing or simulating
complex phenomena, such as rare events triggered by random fluctuations with
heavy as well as light tail statistical features.
