  Operator inference learns low-dimensional dynamical-system models with
polynomial nonlinear terms from trajectories of high-dimensional physical
systems (non-intrusive model reduction). This work focuses on the large class
of physical systems that can be well described by models with quadratic
nonlinear terms and proposes a regularizer for operator inference that induces
a stability bias onto quadratic models. The proposed regularizer is physics
informed in the sense that it penalizes quadratic terms with large norms and so
explicitly leverages the quadratic model form that is given by the underlying
physics. This means that the proposed approach judiciously learns from data and
physical insights combined, rather than from either data or physics alone.
Additionally, a formulation of operator inference is proposed that enforces
model constraints for preserving structure such as symmetry and definiteness in
the linear terms. Numerical results demonstrate that models learned with
operator inference and the proposed regularizer and structure preservation are
accurate and stable even in cases where using no regularization or Tikhonov
regularization leads to models that are unstable.
