Data-Driven Learning for the Mori--Zwanzig Formalism: A Generalization of the Koopman Learning Framework

Abstract

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.