A prioriestimation of memory effects in reduced-order models of nonlinear systems using the Mori–Zwanzig formalism

Abstract

Reduced models of nonlinear dynamical systems require closure, or the modelling of the unresolved modes. The Mori-Zwanzig procedure can be used to derive formally closed evolution equations for the resolved physics. In these equations, the unclosed terms are recast as a memory integral involving the time history of the resolved variables. While this procedure does not reduce the complexity of the original system, these equations can serve as a mathematically consistent basis to develop closures based on memory approximations. In this scenario, knowledge of the memory kernel is paramount in assessing the validity of a memory approximation. Unravelling the memory kernel requires solving the orthogonal dynamics, which is a high-dimensional partial differential equation that is intractable, in general. A method to estimate the memory kernel , using full-order solution snapshots, is proposed. The key idea is to solve a pseudo orthogonal dynamics equation, which has a convenient Liouville form, instead. This ersatz arises from the assumption that the semi-group of the orthogonal dynamics is a composition operator for one observable. The method is exact for linear systems. Numerical results on the Burgers and Kuramoto-Sivashinsky equations demonstrate that the proposed technique can provide valuable information about the memory kernel.

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

“…We note that data‐driven closure modeling for non‐ROM settings is an extremely active research area (see, eg, the works of Duraisamy et al and Ling et al). We also note that there are other DD‐ROM closure models . We emphasize, however, that these DD‐ROM closure models are different from our DDC‐ROM in the following respects.…”

“…We also plan to investigate alternative approaches to determine the optimal value of the parameter m, which is used in (34). In the numerical investigation in Section 4, we chose m by trial and error, aiming at maximizing the accuracy of the ROM energy coefficients.…”

“…We also note that there are other DD-ROM closure models. [30][31][32][33][34][35][36][37] We emphasize, however, that these DD-ROM closure models are different from our DDC-ROM in the following respects.…”

“…For a fair comparison between the DDC-ROM and the CDDC-ROM, for each model, we used optimal parameters m (ie, the parameter used in (34) to ensure the CDDC-ROM's computational efficiency), tol (ie, the tolerance value used in the truncated SVD method for the constrained least squares problem), and (ie, the tolerance used to enforce the constraints A ii ≤ − in the constrained optimization problem solved with the MATLAB toolbox lsqlin with the interior-point method; see Section 4.2). Those optimal parameters were selected to ensure that the ROM energy evolution matches best the FOM energy evolution.…”

Quantum mechanics and data assimilation

“…In particular, in many applications of interest one does have access to a first-principles parametric model (e.g., a numerical weather model), which even if imperfect, may be indispensable in intrinsically high-dimensional applications. At present, we do not have a technique allowing us to seamlessly combine a data-driven Koopman op-erator model with a first-principles state space model, although recent techniques on semiparametric modeling [44] and the Mori-Zwanzig formalism [45] could pave the way for such developments. That being said, it should be noted that in a number of phenomena of interest (e.g., large-scale coherent patterns in climate dynamics such as the El Niño Southern Oscillation and the Madden-Julian Oscillation) there are simply no "perfect" first-principles governing equations, while the effective dimension of the dynamics is moderate.…”