  Reconstructing noise-driven nonlinear networks from time series of output
variables is a challenging problem, which turns to be very difficult when
nonlinearity of dynamics, strong noise impacts and low measurement frequencies
jointly affect. In this Letter, we propose a general method that introduces a
number of nonlinear terms of the measurable variables as artificial and new
variables, and uses the expanded variables to linearize nonlinear differential
equations. Moreover, we use two-time correlations to decompose effects of
system dynamics and noise driving. With these transformations, reconstructing
nonlinear dynamics of the original system is equivalent to solving linear
dynamics of the expanded system at the least squares approximations. We can
well reconstruct nonlinear networks, including all dynamic nonlinearities,
network links, and noise statistical characteristics, as sampling frequency is
rather low. Numerical results fully verify the validity of theoretical
derivations.
