On the estimation of the Mori-Zwanzig memory integral
This work provides theoretical guarantees for reduced-order modeling in statistical physics and dynamical systems, addressing a long-standing need for provably convergent memory approximations.
The authors develop rigorous error bounds and convergence conditions for approximations of the Mori-Zwanzig memory integral, including short-memory, t-model, and hierarchical approximations, and provide numerical validation on linear and nonlinear systems.
We develop rigorous estimates and provably convergent approximations for the memory integral in the Mori-Zwanzig (MZ) formulation. The new theory is built upon rigorous mathematical foundations and is presented for both state-space and probability density function space formulations of the MZ equation. In particular, we derive errors bounds and sufficient convergence conditions for short-memory approximations, the $t$-model, and hierarchical (finite-memory) approximations. In addition, we derive computable upper bounds for the MZ memory integral, which allow us to estimate (a priori) the contribution of the MZ memory to the dynamics. Numerical examples demonstrating convergence of the proposed algorithms are presented for linear and nonlinear dynamical systems evolving from random initial states.