Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations

Khodabakhshi, Parisa; Willcox, Karen E.

doi:10.1016/j.cma.2021.114296

Title: Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations

Abstract

In this paper we present a non-intrusive data-driven approach for model reduction of nonlinear systems. The approach considers the particular case of nonlinear partial differential equations (PDEs) that form systems of partial differential–algebraic equations (PDAEs) when lifted to polynomial form. Such systems arise, for example, when the governing equations include Arrhenius reaction terms (e.g., in reacting flow models) and thermodynamic terms (e.g., the Helmholtz free energy terms in a phase-field solidification model). Using the known structured form of the lifted algebraic equations, the approach computes the reduced operators for the algebraic equations explicitly, using straightforward linear algebra operations on the basis matrices. The reduced operators for the differential equations are inferred from lifted snapshot data using operator inference, which solves a linear least squares regression problem. The approach is illustrated for the nonlinear model of solidification of a pure material. The lifting transformations reformulate the solidification PDEs as a system of PDAEs that have cubic structure. The operators of the lifted system for this solidification example have affine dependence on key process parameters, permitting us to learn a parametric reduced model with operator inference. Numerical experiments show the effectiveness of the resulting reduced models in capturing key aspects of themore » solidification dynamics.« less

Authors:

Khodabakhshi, Parisa ^[1];

^[1]

University of Texas, Austin, TX (United States)

Publication Date:: Sat Nov 27 00:00:00 EST 2021

Research Org.:: Univ. of Texas, Austin, TX (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1976972

Alternate Identifier(s):: OSTI ID: 1840636

Grant/Contract Number:: SC0019303

Resource Type:: Accepted Manuscript

Journal Name:: Computer Methods in Applied Mechanics and Engineering

Additional Journal Information:: Journal Volume: 389; Journal Issue: C; Journal ID: ISSN 0045-7825

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 42 ENGINEERING; 97 MATHEMATICS AND COMPUTING; nonlinear model reduction; lifting transformations; differential–algebraic equation; operator inference; additive manufacturing; solidification

Citation Formats


                    Khodabakhshi, Parisa, and Willcox, Karen E. Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations.  United States: N. p., 2021. 
Web.  doi:10.1016/j.cma.2021.114296.

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                    Khodabakhshi, Parisa, & Willcox, Karen E. Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations.  United States.  https://doi.org/10.1016/j.cma.2021.114296

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                    Khodabakhshi, Parisa, and Willcox, Karen E. Sat .  
"Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations".  United States.  https://doi.org/10.1016/j.cma.2021.114296.  https://www.osti.gov/servlets/purl/1976972.

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@article{osti_1976972,

  title        = {Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations},

  author       = {Khodabakhshi, Parisa and Willcox, Karen E.},

  abstractNote = {In this paper we present a non-intrusive data-driven approach for model reduction of nonlinear systems. The approach considers the particular case of nonlinear partial differential equations (PDEs) that form systems of partial differential–algebraic equations (PDAEs) when lifted to polynomial form. Such systems arise, for example, when the governing equations include Arrhenius reaction terms (e.g., in reacting flow models) and thermodynamic terms (e.g., the Helmholtz free energy terms in a phase-field solidification model). Using the known structured form of the lifted algebraic equations, the approach computes the reduced operators for the algebraic equations explicitly, using straightforward linear algebra operations on the basis matrices. The reduced operators for the differential equations are inferred from lifted snapshot data using operator inference, which solves a linear least squares regression problem. The approach is illustrated for the nonlinear model of solidification of a pure material. The lifting transformations reformulate the solidification PDEs as a system of PDAEs that have cubic structure. The operators of the lifted system for this solidification example have affine dependence on key process parameters, permitting us to learn a parametric reduced model with operator inference. Numerical experiments show the effectiveness of the resulting reduced models in capturing key aspects of the solidification dynamics.},

  doi          = {10.1016/j.cma.2021.114296},

  journal      = {Computer Methods in Applied Mechanics and Engineering},

  number       = C,

  volume       = 389,

  place        = {United States},

  year         = {Sat Nov 27 00:00:00 EST 2021},

  month        = {Sat Nov 27 00:00:00 EST 2021}

}

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Title: Non-intrusive data-driven model reduction for differential–algebraic equations derived from lifting transformations

Abstract

Citation Formats

Modeling and control of physical processes using proper orthogonal decomposition journal, January 2001

A POD-EIM reduced two-scale model for crystal growth journal, July 2014

Missing Point Estimation in Models Described by Proper Orthogonal Decomposition journal, November 2008

Compressed sensing and dynamic mode decomposition journal, January 2015

Data-Driven Reduced Model Construction with Time-Domain Loewner Models journal, January 2017

Model Order Reduction of Differential Algebraic Equations Arising from the Simulation of Gas Transport Networks book, January 2014

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Data-driven operator inference for nonintrusive projection-based model reduction journal, July 2016

An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations journal, November 2004

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Phase-Field Simulation of Solidification journal, August 2002

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An efficient reduced-order modeling approach for non-linear parametrized partial differential equations journal, October 2008

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Two-Sided Projection Methods for Nonlinear Model Order Reduction journal, January 2015

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Data-driven model order reduction of quadratic-bilinear systems: The Loewner framework for QB systems journal, July 2018

POD-based model reduction with empirical interpolation applied to nonlinear elasticity: POD WITH EMPIRICAL INTERPOLATION APPLIED TO NONLINEAR ELASTICITY journal, December 2015

Projection-based model reduction: Formulations for physics-based machine learning journal, January 2019

Phase Field Simulations of Microstructure Evolution in IN718 using a Surrogate Ni–Fe–Nb Alloy during Laser Powder Bed Fusion journal, December 2018

Non-linear model reduction for the Navier–Stokes equations using residual DEIM method journal, April 2014

Data-Driven Parametrized Model Reduction in the Loewner Framework journal, January 2014

Analysis of Fluid Flows via Spectral Properties of the Koopman Operator journal, January 2013

QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems journal, September 2011

Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator journal, January 2017

The phase-field method: simulation of alloy dendritic solidification during recalescence journal, March 1996

Model Reduction of Descriptor Systems by Interpolatory Projection Methods journal, January 2013

Model Reduction for Transport-Dominated Problems via Online Adaptive Bases and Adaptive Sampling journal, January 2020

Reduced-order modeling of parameterized PDEs using time-space-parameter principal component analysis journal, November 2009

Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms journal, December 2020

Greedy Nonintrusive Reduced Order Model for Fluid Dynamics journal, December 2018

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems journal, January 2015

Turbulence and the dynamics of coherent structures. I. Coherent structures journal, January 1987

Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition journal, June 2019

Learning physics-based models from data: perspectives from inverse problems and model reduction journal, May 2021

Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control journal, July 2018

Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media journal, August 2011

Recasting nonlinear differential equations as S-systems: a canonical nonlinear form journal, November 1987

Nonlinear Model Reduction via Discrete Empirical Interpolation journal, January 2010

Phase-field model for isothermal phase transitions in binary alloys journal, May 1992

Learning Physics-Based Reduced-Order Models for a Single-Injector Combustion Process journal, June 2020

Modeling and control of physical processes using proper orthogonal decomposition
journal, January 2001

A POD-EIM reduced two-scale model for crystal growth
journal, July 2014

Missing Point Estimation in Models Described by Proper Orthogonal Decomposition
journal, November 2008

Compressed sensing and dynamic mode decomposition
journal, January 2015

Data-Driven Reduced Model Construction with Time-Domain Loewner Models
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Model Order Reduction of Differential Algebraic Equations Arising from the Simulation of Gas Transport Networks
book, January 2014

Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems
journal, May 2020

Dynamic mode decomposition of numerical and experimental data
journal, July 2010

POD model order reduction with space-adapted snapshots for incompressible flows
journal, September 2019

Data-driven operator inference for nonintrusive projection-based model reduction
journal, July 2016

An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations
journal, November 2004

Development and application of reduced-order modeling procedures for subsurface flow simulation
journal, February 2009

Phase-Field Simulation of Solidification
journal, August 2002

Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures
journal, January 2016

Data-driven reduced-order models via regularised Operator Inference for a single-injector combustion process
journal, January 2021

An efficient reduced-order modeling approach for non-linear parametrized partial differential equations
journal, October 2008

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journal, January 2021

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journal, June 2016

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journal, January 2015

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journal, January 2008

Data-driven model order reduction of quadratic-bilinear systems: The Loewner framework for QB systems
journal, July 2018

POD-based model reduction with empirical interpolation applied to nonlinear elasticity: POD WITH EMPIRICAL INTERPOLATION APPLIED TO NONLINEAR ELASTICITY
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Projection-based model reduction: Formulations for physics-based machine learning
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journal, April 2014

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journal, January 2014

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journal, January 2013

QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems
journal, September 2011

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journal, January 2017

The phase-field method: simulation of alloy dendritic solidification during recalescence
journal, March 1996

Model Reduction of Descriptor Systems by Interpolatory Projection Methods
journal, January 2013

Model Reduction for Transport-Dominated Problems via Online Adaptive Bases and Adaptive Sampling
journal, January 2020

Reduced-order modeling of parameterized PDEs using time-space-parameter principal component analysis
journal, November 2009

Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms
journal, December 2020

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journal, December 2018

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems
journal, January 2015

Turbulence and the dynamics of coherent structures. I. Coherent structures
journal, January 1987

Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition
journal, June 2019

Learning physics-based models from data: perspectives from inverse problems and model reduction
journal, May 2021

Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
journal, July 2018

Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media
journal, August 2011

Recasting nonlinear differential equations as S-systems: a canonical nonlinear form
journal, November 1987

Nonlinear Model Reduction via Discrete Empirical Interpolation
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Phase-field model for isothermal phase transitions in binary alloys
journal, May 1992

Learning Physics-Based Reduced-Order Models for a Single-Injector Combustion Process
journal, June 2020