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 \emph{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：模型降阶技术通过构建低复杂度高保真度的代理模型，能够快速精确地求解参数化微分方程。针对参数化非线性哈密顿系统开发降阶模型面临多重挑战：(i) 编码动力学物理属性的几何结构；(ii) 保守动力学缓慢衰减的Kolmogorov $n$-宽度；(iii) 非线性流速的梯度结构；(iv) 系统状态数值秩随时间与参数的高变化性。本文提出通过结构保持自适应方法应对这些挑战，该方法将辛动态低秩逼近与自适应梯度保持超降阶及参数采样相结合。此外，我们提出通过监测与哈密顿向量场投影误差相关的误差指示器，动态调整降阶基空间和超降阶空间的维度。所得自适应超降阶模型能保持哈密顿流的几何结构，无需预先获取动力学信息，且求解成本与全阶模型维度及测试参数数量均呈线性关系。数值实验表明，完全自适应模型相较于原始模型与常规降阶模型具有显著的性能提升。