AI 中文总结
本文提出一种与系统无关、无需优化的二次投影矩阵构造框架，结合非线性矩匹配与二次流形，在输运主导基准问题中实现高保真轨迹重构并节省在线计算量。
AI 中文摘要
基于二次流形的模型降阶为规避线性子空间对Kolmogorov n-宽度衰减缓慢的线性控制系统的局限性提供了可行途径，但文献中仍缺乏构建此类二次近似的系统理论框架。本文提出一种与系统无关、无需优化的框架，用于直接构造二次投影矩阵。我们证明，综合得到的降阶模型可匹配全阶系统的非线性矩并保留其精确中心流形映射，从而确保在特定输入类下对稳态输出的渐近跟踪。针对输运主导的基准问题，即一维阻尼波动方程和对流方程的数值结果表明，所提框架在显著降维的状态空间内实现了高保真轨迹重构，带来了可观的在线计算节省。
英文摘要
Quadratic manifold-based model order reduction offers a viable pathway to circumvent the limitations of linear subspaces for linear control systems characterized by slow Kolmogorov $n$-width decay. However, a system-theoretic framework for constructing such quadratic approximations remains absent from the literature. This paper presents a system-agnostic, optimization-free framework for the direct construction of quadratic projection matrices. We prove that the synthesized reduced-order model matches the nonlinear moments of the full-order system and preserves its exact center manifold mapping, thereby ensuring asymptotic tracking of steady-state outputs under specific input classes. Numerical results on transport-dominated benchmark problems, namely, the one-dimensional damped wave and advection equations, show that the proposed framework achieves high-fidelity trajectory reconstruction within a significantly reduced-dimensional state space, yielding substantial online computational savings.
