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Published May 2020 | Accepted Version
Journal Article Open

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


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.

Additional Information

© 2020 Elsevier. Received 5 December 2019, Revised 30 January 2020, Accepted 7 February 2020, Available online 19 February 2020, Version of Record 5 March 2020. 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. CRediT authorship contribution statement. Elizabeth Qian: Conceptualization, Methodology, Software, Formal analysis, Validation, Writing - original draft, Visualization. Boris Kramer: Conceptualization, Methodology, Software, Writing - reviewing & editing. Benjamin Peherstorfer: Conceptualization, Methodology, Formal analysis, Writing - reviewing & editing. Karen Willcox: Conceptualization, Methodology, Resources, Writing - reviewing & editing, Supervision, Funding acquisition.

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Accepted Version - 1912.08177.pdf


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