  Stochastic differential equations (SDEs) are one of the most important
representations of dynamical systems. They are notable for the ability to
include a deterministic component of the system and a stochastic one to
represent random unknown factors. However, this makes learning SDEs much more
challenging than ordinary differential equations (ODEs). In this paper, we
propose a data driven approach where parameters of the SDE are represented by a
neural network with a built-in SDE integration scheme. The loss function is
based on a maximum likelihood criterion, under order one Markov Gaussian
assumptions. The algorithm is applied to the geometric brownian motion and a
stochastic version of the Lorenz-63 model. The latter is particularly hard to
handle due to the presence of a stochastic component that depends on the state.
The algorithm performance is attested using different simulations results.
Besides, comparisons are performed with the reference gradient matching method
used for non linear drift estimation, and a neural networks-based method, that
does not consider the stochastic term.
