  A theoretical framework which unifies the conventional Mori--Zwanzig
formalism and the approximate Koopman learning is presented. In this framework,
the Mori--Zwanzig formalism, developed in statistical mechanics to tackle the
hard problem of construction of reduced-order dynamics for high-dimensional
dynamical systems, can be considered as a natural generalization of the Koopman
description of the dynamical system. We next show that similar to the
approximate Koopman learning methods, data-driven methods can be developed for
the Mori--Zwanzig formalism with Mori's linear projection operator. We
developed two algorithms to extract the key operators, the Markov and the
memory kernel, using time series of a reduced set of observables in a dynamical
system. We adopted the Lorenz `96 system as a test problem and solved for the
operators, which exhibit complex behaviors which are unlikely to be captured by
traditional modeling approaches, in Mori--Zwanzig analysis. The nontrivial
Generalized Fluctuation Dissipation relationship, which relates the memory
kernel with the two-time correlation statistics of the orthogonal dynamics, was
numerically verified as a validation of the solved operators. We present
numerical evidence that the Generalized Langevin Equation, a key construct in
the Mori--Zwanzig formalism, is more advantageous in predicting the evolution
of the reduced set of observables than the conventional approximate Koopman
operators.
