Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which emphasizes the geometric nature of the objects involved. The crucial ingredient is the construction of an embedding for the low-dimensional submanifold and a compatible reduction map, for which we discuss several options. Our general framework allows capturing and generalizing several existing MOR techniques, such as structure preservation for Lagrangian- or Hamiltonian dynamics, and using nonlinear projections that are, for instance, relevant in transport-dominated problems. The joint abstraction can be used to derive shared theoretical properties for different methods, such as an exact reproduction result. To connect our framework to existing work in the field, we demonstrate that various techniques for data-driven construction of nonlinear projections can be included in our framework.

翻译：利用非线性投影并保持模型降阶（MOR）中的结构是当前活跃的研究领域。本文针对光滑流形上的模型降阶提出了一种新颖的微分几何框架，该框架强调了所涉对象的几何本质。关键要素在于低维子流形的嵌入构造及其相容的降阶映射，我们对此讨论了多种方案。所提出的通用框架能够概括并推广多种现有MOR技术，例如拉格朗日或哈密顿动力学的结构保持，以及针对输运主导问题中具有重要意义的非线性投影方法。该联合抽象可推导不同方法共有的理论特性，如精确重构结果。为连接本框架与现有研究成果，我们证明了多种数据驱动型非线性投影构建技术均可纳入本框架。