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Title: Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems

Abstract

In this work, we present Lift & Learn, a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system’s governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh–Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model.

Authors:
 [1];  [2];  [3];  [4]
+ Show Author Affiliations
  1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
  2. Univ. of California, San Diego, CA (United States)
  3. New York Univ. (NYU), NY (United States). Courant Institute of Mathematical Sciences
  4. Univ. of Texas, Austin, TX (United States). Oden Institute for Computational Engineering and Sciences
Publication Date:
Research Org.:
Univ. of Texas, Austin, TX (United States); New York Univ. (NYU), NY (United States)
Sponsoring Org.:
USDOE Office of Science (SC); US Air Force Office of Scientific Research (AFOSR)
OSTI Identifier:
1803677
Alternate Identifier(s):
OSTI ID: 1603700
Grant/Contract Number:  
SC0019303; SC0019334; FA9550-17-1-0195; FA9550-15-1-0038; FA9550-18-1-0023
Resource Type:
Accepted Manuscript
Journal Name:
Physica. D, Nonlinear Phenomena
Additional Journal Information:
Journal Volume: 406; Journal Issue: C; Journal ID: ISSN 0167-2789
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Data-driven model reduction; Scientific machine learning; Dynamical systems; Partial differential equations; Lifting map

Citation Formats

Qian, Elizabeth, Kramer, Boris, Peherstorfer, Benjamin, and Willcox, Karen. Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems. United States: N. p., 2020. Web. doi:10.1016/j.physd.2020.132401.
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Qian, Elizabeth, Kramer, Boris, Peherstorfer, Benjamin, & Willcox, Karen. Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems. United States. https://doi.org/10.1016/j.physd.2020.132401
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Qian, Elizabeth, Kramer, Boris, Peherstorfer, Benjamin, and Willcox, Karen. Wed . "Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems". United States. https://doi.org/10.1016/j.physd.2020.132401. https://www.osti.gov/servlets/purl/1803677.
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@article{osti_1803677,
title = {Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems},
author = {Qian, Elizabeth and Kramer, Boris and Peherstorfer, Benjamin and Willcox, Karen},
abstractNote = {In this work, we present Lift & Learn, a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system’s governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh–Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model.},
doi = {10.1016/j.physd.2020.132401},
journal = {Physica. D, Nonlinear Phenomena},
number = C,
volume = 406,
place = {United States},
year = {Wed Feb 19 00:00:00 EST 2020},
month = {Wed Feb 19 00:00:00 EST 2020}
}
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https://doi.org/10.1016/j.physd.2020.132401
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