Renormalized Reduced Order Models with Memory for Long Time Prediction
This work provides a method for stable and accurate reduced-order modeling in complex dynamical systems without timescale separation, which is a known bottleneck in model reduction.
The authors address the challenge of constructing reduced models for long-time prediction in systems with two-way transfer of activity between resolved and unresolved variables. They stabilize the Mori-Zwanzig reduced models using dynamic information from the full system, achieving accurate long-time predictions for the Korteweg-de Vries equation.
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.