  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.
