  Reduced Order Models (ROMs) of complex, nonlinear dynamical systems often
require closure, which is the process of representing the contribution of the
unresolved physics on the resolved physics. The Mori-Zwanzig (M-Z) procedure
allows one to write down formally closed evolution equations for the resolved
physics. In these equations, the unclosed terms are recast as a memory integral
involving the past history of the resolved variables, and a "noise" term. While
the M-Z procedure does not directly reduce the complexity of the original
system, these equations can serve as a mathematically consistent starting point
to develop closures based on approximations of the memory. In this scenario, a
priori knowledge of the memory kernel, which is not explicitly known for
nonlinear systems, is of paramount importance to assess the validity of a
memory approximation. Unraveling the memory kernel requires the determination
of the orthogonal dynamics which is a projected high-dimensional partial
differential equation that is not tractable in general. A method to estimate
the memory kernel a priori, using full-order solution snapshots, is proposed.
The main idea is to solve a pseudo orthogonal dynamics equation, that has a
convenient Liouville form, instead of the original one. This ersatz arises from
the assumption that the semi-group of the orthogonal dynamics operator is a
composition operator, akin to semi-groups of Liouville operators, for one
observable. The method is exact in the linear case where the kernel is known
explicitly. Results for under-resolved simulations of the Burgers and
Kuramoto-Sivashinsky equations demonstrate that the proposed technique can
accurately reconstruct the transfer of information between the resolved and
unresolved dynamics through memory, and provide valuable information about the
kernel.
