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 still 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 resulting fully adaptive models compared to the original and reduced order models.

翻译：模型降阶通过构建低复杂度高保真替代模型，可快速精确求解参数化微分方程。然而，参数非线性哈密顿系统的降阶模型开发仍面临多重挑战：(i) 编码动力学物理特性的几何结构保持问题；(ii) 保守动力学中缓慢衰减的Kolmogorov $n$-宽度问题；(iii) 非线性流速度的梯度结构问题；(iv) 状态数值秩随时间和参数的高波动性问题。本文提出一种保结构自适应方法处理上述问题，该方法融合辛动态低秩逼近、自适应梯度保持超降阶及参数采样技术。此外，我们通过哈密顿向量场投影误差相关的误差指示器监测降阶解质量，动态调整降基空间与超降阶空间的维度。由此得到的自适应超降阶模型既保持了哈密顿流的几何结构，又无需依赖动力学先验信息，其求解复杂度与全阶模型维度和测试参数数量呈线性关系。数值实验表明，与原始模型及常规降阶模型相比，所提出的全自适应模型具有更优性能。