  Many physical systems characterized by nonlinear multiscale interactions can
be effectively modeled by treating unresolved degrees of freedom as random
fluctuations. However, even when the microscopic governing equations and
qualitative macroscopic behavior are known, it is often difficult to derive a
stochastic model that is consistent with observations. This is especially true
for systems such as turbulence where the perturbations do not behave like
Gaussian white noise, introducing non-Markovian behavior to the dynamics. We
address these challenges with a framework for identifying interpretable
stochastic nonlinear dynamics from experimental data, using both forward and
adjoint Fokker-Planck equations to enforce statistical consistency. If the form
of the Langevin equation is unknown, a simple sparsifying procedure can provide
an appropriate functional form. We demonstrate that this method can effectively
learn stochastic models in two artificial examples: recovering a nonlinear
Langevin equation forced by colored noise and approximating the second-order
dynamics of a particle in a double-well potential with the corresponding
first-order bifurcation normal form. Finally, we apply the proposed method to
experimental measurements of a turbulent bluff body wake and show that the
statistical behavior of the center of pressure can be described by the dynamics
of the corresponding laminar flow driven by nonlinear state-dependent noise.
