Physics constrained nonlinear regression models for time series

Andrew J. Majda, John Harlim

Research output: Contribution to journalArticle

Abstract

A central issue in contemporary science is the development of data driven statistical nonlinear dynamical models for time series of partial observations of nature or a complex physical model. It has been established recently that ad hoc quadratic multi-level regression (MLR) models can have finite-time blow up of statistical solutions and/or pathological behaviour of their invariant measure. Here a new class of physics constrained multi-level quadratic regression models are introduced, analysed and applied to build reduced stochastic models from data of nonlinear systems. These models have the advantages of incorporating memory effects in time as well as the nonlinear noise from energy conserving nonlinear interactions. The mathematical guidelines for the performance and behaviour of these physics constrained MLR models as well as filtering algorithms for their implementation are developed here. Data driven applications of these new multi-level nonlinear regression models are developed for test models involving a nonlinear oscillator with memory effects and the difficult test case of the truncated Burgers-Hopf model. These new physics constrained quadratic MLR models are proposed here as process models for Bayesian estimation through Markov chain Monte Carlo algorithms of low frequency behaviour in complex physical data.

Original languageEnglish (US)
Article number201
Pages (from-to)201-217
Number of pages17
JournalNonlinearity
Volume26
Issue number1
DOIs
StatePublished - Jan 2013

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Nonlinear Regression Model
Multilevel Models
regression analysis
Time series
Regression Model
Physics
physics
Memory Effect
Data-driven
Statistical Solutions
Partial Observation
Finite Time Blow-up
Markov Chain Monte Carlo Algorithms
Nonlinear Interaction
Reduced Model
Nonlinear Oscillator
Bayesian Estimation
Dynamical Model
Physical Model
Invariant Measure

ASJC Scopus subject areas

  • Applied Mathematics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Physics constrained nonlinear regression models for time series. / Majda, Andrew J.; Harlim, John.

In: Nonlinearity, Vol. 26, No. 1, 201, 01.2013, p. 201-217.

Research output: Contribution to journalArticle

Majda, Andrew J. ; Harlim, John. / Physics constrained nonlinear regression models for time series. In: Nonlinearity. 2013 ; Vol. 26, No. 1. pp. 201-217.
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