TY - JOUR

T1 - Lift & Learn

T2 - Physics-informed machine learning for large-scale nonlinear dynamical systems

AU - Qian, Elizabeth

AU - Kramer, Boris

AU - Peherstorfer, Benjamin

AU - Willcox, Karen

N1 - Funding Information:
This work was supported in part by the US Air Force Center of Excellence on Multi-Fidelity Modeling of Rocket Combustor Dynamics award FA9550-17-1-0195 , the Air Force Office of Scientific Research MURI on managing multiple information sources of multi-physics systems awards FA9550-15-1-0038 and FA9550-18-1-0023 , the US Department of Energy Applied Mathematics MMICC Program award DE-SC0019303 , and the SUTD-MIT International Design Centre, United States of America . The first author also acknowledges support from the National Science Foundation Graduate Research Fellowship Program and the Fannie and John Hertz Foundation, United States of America . The third author was partially supported by the US Department of Energy , Office of Advanced Scientific Computing Research , Applied Mathematics Program (Program Manager Dr. Steven Lee), DOE Award DESC0019334 .
Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/5

Y1 - 2020/5

N2 - 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.

AB - 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.

KW - Data-driven model reduction

KW - Dynamical systems

KW - Lifting map

KW - Partial differential equations

KW - Scientific machine learning

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U2 - 10.1016/j.physd.2020.132401

DO - 10.1016/j.physd.2020.132401

M3 - Article

AN - SCOPUS:85081003697

SN - 0167-2789

VL - 406

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

M1 - 132401

ER -