Fundamentallimitations of Ad hoc linear and quadratic multi-level regression models forphysical systems

A central issue in contemporary applied mathematics is the development ofsimpler dynamical models for a reduced subset of variables in complex highdimensional dynamical systems with many spatio-temporal scales. Recently, adhoc quadratic multi-level regression models have been proposed to providesuitable reduced nonlinear models directly from data. The main resultsdeveloped here are rigorous theorems demonstrating the non-physical finitetime blow-up and large time instability in statistical solutions of generalscalar multi-level quadratic regression models with corresponding unphysicalfeatures of the invariant measure. Surprising intrinsic model errors due todiscrete sampling errors are also shown to occur rigorously even for linearmulti-level regression dynamic models. all of these theoretical results arecorroborated by numerical experiments with simple models. Single levelnonlinear regression strategies with physical cubic damping are shown to havesignificant skill on the same test problems.