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, linear basis approximations can suffer from slowly decaying Kolmogorov $N$-width, especially in wave-type problems, which then requires a large basis size. 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 to 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：本研究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约简的新方法。经典辛模型约简方法采用线性辛子空间，在降维坐标系中表示高维系统状态。虽然此类近似保留了哈密顿系统的辛结构，但线性基近似可能因柯尔莫哥洛夫$N$-宽度衰减缓慢（尤其在波动类问题中）而需采用大规模基组。我们基于近期发展的二次流形提出了两种模型约简方法，各具优势与局限。作为方法论核心的状态近似中引入二次项，使我们能更有效地表征问题本身固有的低维特性。两种方法在远超训练数据范围的场景中均能有效进行预测，同时提供比线性辛降阶模型更精确的解。