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，该框架强调所涉及对象的几何本质。其关键要素是构建低维子流形的嵌入及兼容的降阶映射，对此我们讨论了多种选择方案。该通用框架能够涵盖并推广多种现有模型降阶技术，例如拉格朗日或哈密顿动力学的结构保持，以及适用于输运主导问题中的非线性投影。通过这一联合抽象，可推导出不同方法共有的理论性质（如精确复现结果）。为连接本框架与领域现有工作，我们证明了多种数据驱动非线性投影构造技术均可纳入该框架。