Fundamental limitations of ad hoc linear and quadratic multi-level regression models for physical systems

Andrew J. Majda, Yuan Yuan

Research output: Contribution to journalArticlepeer-review


A central issue in contemporary applied mathematics is the development of simpler dynamical models for a reduced subset of variables in complex high dimensional dynamical systems with many spatio-temporal scales. Recently, ad hoc quadratic multi-level regression models have been proposed to provide suitable reduced nonlinear models directly from data. The main results developed here are rigorous theorems demonstrating the non-physical finite time blow-up and large time instability in statistical solutions of general scalar multi-level quadratic regression models with corresponding unphysical features of the invariant measure. Surprising intrinsic model errors due to discrete sampling errors are also shown to occur rigorously even for linear multi-level regression dynamic models. all of these theoretical results are corroborated by numerical experiments with simple models. Single level nonlinear regression strategies with physical cubic damping are shown to have significant skill on the same test problems.

Original languageEnglish (US)
Pages (from-to)1333-1363
Number of pages31
JournalDiscrete and Continuous Dynamical Systems - Series B
Issue number4
StatePublished - Jun 2012


  • Instability
  • Model error
  • Nonlinear regression models
  • Unphysical blow up

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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