  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.
