FUNDAMENTAL LIMITATIONS OF AD HOC LINEAR AND QUADRATIC MULTI-LEVEL REGRESSION MODELS FOR PHYSICAL SYSTEMS

By Andrew J. Majda, Yuan Yuan. 2012

47 citations recorded by the source corpus.

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.

Subfields: Control and Systems Engineering, Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

Theory-Guided Data Science: A New Paradigm for Scientific Discovery from Data (2016, background): “driven PDEs can be found in a recent work in climate science [55], [56], where physically constrained time-series regression models were developed to incorporate memory”
Toward enhanced understanding and projections of climate extremes using physics-guided data mining techniques (2014, methodology): “Data-driven methods may be complementary to physics and may need to be constrained by physics (e.g., see Majda and Yuan, 2012; Majda and Hardin, 2013).”
A Big Data Guide to Understanding Climate Change: The Case for Theory-Guided Data Science (2014, unknown): “Finally, within predictive models, it has been shown both theoretically 35 and in practice 36 that theory-agnostic models generalize poorly to unseen data despite being highly accurate on observed data.”
Challenges in Climate Science and Contemporary Applied Mathematics (2012, background): “Ad-hoc nonlinear regression models can exhibit improved skill in a training time series (see references in [100]) but can suffer unphysical finite-time blow-up of statistical solutions, as well as unphysical pathology in their invariant measure [100].”
Physics constrained nonlinear regression models for time series (2012, background): “Such nonuniqueness occurs even for linear regression models and explicit examples are given in [15].”