  This work presents two novel approaches for the symplectic model reduction of
high-dimensional Hamiltonian systems using data-driven quadratic manifolds.
Classical symplectic model reduction approaches employ linear symplectic
subspaces for representing the high-dimensional system states in a
reduced-dimensional coordinate system. While these approximations respect the
symplectic nature of Hamiltonian systems, linear basis approximations can
suffer from slowly decaying Kolmogorov $N$-width, especially in wave-type
problems, which then requires a large basis size. We propose two different
model reduction methods based on recently developed quadratic manifolds, each
presenting its own advantages and limitations. The addition of quadratic terms
to the state approximation, which sits at the heart of the proposed
methodologies, enables us to better represent intrinsic low-dimensionality in
the problem at hand. Both approaches are effective for issuing predictions in
settings well outside the range of their training data while providing more
accurate solutions than the linear symplectic reduced-order models.
