  We present a data-driven approach to determine the memory kernel and random
noise in generalized Langevin equations. To facilitate practical
implementations, we parameterize the kernel function in the Laplace domain by a
rational function, with coefficients directly linked to the equilibrium
statistics of the coarse-grain variables. We show that such an approximation
can be constructed to arbitrarily high order and the resulting generalized
Langevin dynamics can be embedded in an extended stochastic model without
explicit memory. We demonstrate how to introduce the stochastic noise so that
the second fluctuation-dissipation theorem is exactly satisfied. Results from
several numerical tests are presented to demonstrate the effectiveness of the
proposed method.
