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

Elizabeth Qian, Boris Kramer, Benjamin Peherstorfer, Karen Willcox

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number132401
JournalPhysica D: Nonlinear Phenomena
StatePublished - May 2020


  • Data-driven model reduction
  • Dynamical systems
  • Lifting map
  • Partial differential equations
  • Scientific machine learning

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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