Renormalized Reduced Order Models with Memory for Long Time Prediction

TL;DR

This paper develops a renormalized Mori-Zwanzig reduced model with memory for long-term prediction of systems with bidirectional activity transfer, stabilizing the model and maintaining accuracy over extended periods.

Contribution

It introduces a novel renormalization approach using dynamic information to stabilize and improve the long-term accuracy of reduced models with memory.

Findings

Renormalized models remain stable over long times.

Models accurately predict system behavior without timescale separation.

Coefficients show algebraic time dependence and incomplete similarity.

Abstract

We examine the challenging problem of constructing reduced models for the long time prediction of systems where there is no timescale separation between the resolved and unresolved variables. In previous work we focused on the case where there was only transfer of activity (e.g. energy, mass) from the resolved to the unresolved variables. Here we investigate the much more difficult case where there is two-way transfer of activity between the resolved and unresolved variables. Like in the case of activity drain out of the resolved variables, even if one starts with an exact formalism, like the Mori-Zwanzig (MZ) formalism, the constructed reduced models can become unstable. We show how to remedy this situation by using dynamic information from the full system to renormalize the MZ reduced models. In addition to being stabilized, the renormalized models can be accurate for very long times. We use the Korteweg-de Vries equation to illustrate the approach. The coefficients of the renormalized models exhibit rich structure, including algebraic time dependence and incomplete similarity.