  Learning unknown stochastic differential equations (SDEs) from observed data
is a significant and challenging task with applications in various fields.
Current approaches often use neural networks to represent drift and diffusion
functions, and construct likelihood-based loss by approximating the transition
density to train these networks. However, these methods often rely on one-step
stochastic numerical schemes, necessitating data with sufficiently high time
resolution. In this paper, we introduce novel approximations to the transition
density of the parameterized SDE: a Gaussian density approximation inspired by
the random perturbation theory of dynamical systems, and its extension, the
dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust
density approximation, our method exhibits superior accuracy compared to
baseline methods in learning the fully unknown drift and diffusion functions
and computing the invariant distribution from trajectory data. And it is
capable of handling trajectory data with low time resolution and variable, even
uncontrollable, time step sizes, such as data generated from Gillespie's
stochastic simulations. We then conduct several experiments across various
scenarios to verify the advantages and robustness of the proposed method.
