  This paper focuses on a stochastic system identification problem: given time
series observations of a stochastic differential equation (SDE) driven by
L\'{e}vy $\alpha$-stable noise, estimate the SDE's drift field. For $\alpha$ in
the interval $[1,2)$, the noise is heavy-tailed, leading to computational
difficulties for methods that compute transition densities and/or likelihoods
in physical space. We propose a Fourier space approach that centers on
computing time-dependent characteristic functions, i.e., Fourier transforms of
time-dependent densities. Parameterizing the unknown drift field using Fourier
series, we formulate a loss consisting of the squared error between predicted
and empirical characteristic functions. We minimize this loss with gradients
computed via the adjoint method. For a variety of one- and two-dimensional
problems, we demonstrate that this method is capable of learning drift fields
in qualitative and/or quantitative agreement with ground truth fields.
