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, the linearity of the approximation imposes a fundamental limitation to the accuracy that can be achieved. 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 in 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.

翻译：本文提出了两种利用数据驱动二次流形对高维哈密顿系统进行辛模型降阶的新方法。经典辛模型降阶方法采用线性辛子空间将高维系统状态表示为降维坐标系。尽管这些近似能保持哈密顿系统的辛特性，但近似的线性特性从根本上限制了可达精度。我们基于近期发展的二次流形提出了两种不同的模型降阶方法，各具优势与局限。作为所提方法论核心的状态近似中添加二次项，使我们能更好地表征问题内在的低维性。两种方法均能在训练数据范围之外的设定中有效进行预测，同时相比线性辛降阶模型提供更精确的解。