  We present a novel way of constructing reduced models for systems of ordinary
differential equations. The reduced models we construct depend on coefficients
which measure the importance of the different terms appearing in the model and
need to be estimated. The proposed approach allows the estimation of these
coefficients on the fly by enforcing the equality of integral quantities of the
solution as computed from the original system and the reduced model. In
particular, the approach combines the concepts of renormalization and effective
field theory developed in the context of high energy physics and the
Mori-Zwanzig formalism of irreversible statistical mechanics. It allows to
construct stable reduced models of higher order than was previously possible.
The method is applied to the problem of computing reduced models for ordinary
differential equation systems resulting from Fourier expansions of singular (or
near-singular) time-dependent partial differential equations. Results for the
1D Burgers and the 3D incompressible Euler equations are used to illustrate the
construction. We also present, for the 1D Burgers and the 3D Euler equations, a
simple and efficient recursive algorithm for calculating the higher order
terms.
