### 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 language | English (US) |
---|---|

Article number | 201 |

Pages (from-to) | 201-217 |

Number of pages | 17 |

Journal | Nonlinearity |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2013 |

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### ASJC Scopus subject areas

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

### Cite this

*Nonlinearity*,

*26*(1), 201-217. [201]. https://doi.org/10.1088/0951-7715/26/1/201

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

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 26, no. 1, 201, pp. 201-217. https://doi.org/10.1088/0951-7715/26/1/201

}

TY - JOUR

T1 - Physics constrained nonlinear regression models for time series

AU - Majda, Andrew J.

AU - Harlim, John

PY - 2013/1

Y1 - 2013/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84870475304&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870475304&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/26/1/201

DO - 10.1088/0951-7715/26/1/201

M3 - Article

AN - SCOPUS:84870475304

VL - 26

SP - 201

EP - 217

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 1

M1 - 201

ER -