  While model order reduction is a promising approach in dealing with
multi-scale time-dependent systems that are too large or too expensive to
simulate for long times, the resulting reduced order models can suffer from
instabilities. We have recently developed a time-dependent renormalization
approach to stabilize such reduced models. In the current work, we extend this
framework by introducing a parameter that controls the time-decay of the memory
of such models and optimally selecting this parameter based on limited fully
resolved simulations. First, we demonstrate our framework on the inviscid
Burgers equation whose solution develops a finite-time singularity. Our
renormalized reduced order models are stable and accurate for long times while
using for their calibration only data from a full order simulation before the
occurrence of the singularity. Furthermore, we apply this framework to the 3D
Euler equations of incompressible fluid flow, where the problem of finite-time
singularity formation is still open and where brute force simulation is only
feasible for short times. Our approach allows us to obtain for the first time a
perturbatively renormalizable model which is stable for long times and includes
all the complex effects present in the 3D Euler dynamics. We find that, in each
application, the renormalization coefficients display algebraic decay with
increasing resolution, and that the parameter which controls the time-decay of
the memory is problem-dependent.
