  Model order reduction provides low-complexity high-fidelity surrogate models
that allow rapid and accurate solutions of parametric differential equations.
The development of reduced order models for parametric nonlinear Hamiltonian
systems is challenged by several factors: (i) the geometric structure encoding
the physical properties of the dynamics; (ii) the slowly decaying Kolmogorov
n-width of conservative dynamics; (iii) the gradient structure of the nonlinear
flow velocity; (iv) high variations in the numerical rank of the state as a
function of time and parameters. We propose to address these aspects via a
structure-preserving adaptive approach that combines symplectic dynamical
low-rank approximation with adaptive gradient-preserving hyper-reduction and
parameters sampling. Additionally, we propose to vary in time the dimensions of
both the reduced basis space and the hyper-reduction space by monitoring the
quality of the reduced solution via an error indicator related to the
projection error of the Hamiltonian vector field. The resulting adaptive
hyper-reduced models preserve the geometric structure of the Hamiltonian flow,
do not rely on prior information on the dynamics, and can be solved at a cost
that is linear in the dimension of the full order model and linear in the
number of test parameters. Numerical experiments demonstrate the improved
performances of the fully adaptive models compared to the original and reduced
models.
