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.

翻译：模型降阶提供了低复杂度高保真的代理模型，能够快速精确地求解参数化微分方程。参数化非线性哈密顿系统的降阶模型开发面临多重挑战：（i）编码动力学物理性质的几何结构；（ii）保守动力学的缓慢衰减Kolmogorov n-宽度；（iii）非线性流速的梯度结构；（iv）状态数值秩随时间和参数变化的高度波动。我们提出采用保持结构的自适应方法解决这些问题，该方法结合了辛动力低秩近似与自适应梯度保持超约简及参数采样。此外，通过监测与哈密顿向量场投影误差相关的误差指标，我们建议在时间上自适应变化约简基空间和超约简空间的维度。所得到的自适应超约简模型保持了哈密顿流的几何结构，无需依赖动力学先验信息，且计算成本与全阶模型维度及测试参数数量呈线性关系。数值实验表明，全自适应模型相比原始模型和约简模型具有更优性能。