Colloquium

The Mori--Zwanzig formalism was developed in statistical mechanics to tackle the hard problem of construction of reduced-order dynamics for high-dimensional dynamical systems. I will first illustrate that Mori-Zwanzig formalism is a natural generalization of the Koopman representation of under-resolved dynamical systems. We then 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 numerically solved operators. Finally, 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. I will present the application of our numerical methods on a Direct Numerical Simulation dataset of a three-dimensional isotropic turbulence problem, if time permits.

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