  We present a new method to approximate the Mori-Zwanzig (MZ) memory integral
in generalized Langevin equations (GLEs) describing the evolution of smooth
observables in high-dimensional nonlinear systems with local interactions.
Building upon the Faber operator series we recently developed for the
orthogonal dynamics propagator, and an exact combinatorial algorithm that
allows us to compute memory kernels from first principles, we demonstrate that
the proposed method is effective in computing auto-correlation functions,
intermediate scattering functions and other important statistical properties of
the observable. We also develop a new stochastic process representation of the
MZ fluctuation term for systems in statistical equilibrium. Numerical
applications are presented for the Fermi-Pasta-Ulam model, and for random wave
propagation in homogeneous media.
