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：利用非线性投影和保持结构进行模型降阶（MOR）是当前活跃的研究领域。本文提出了一种面向光滑流形上模型降阶的新型微分几何框架，该框架强调所涉及对象的几何本质。关键要素在于构建低维子流形的嵌入及相容的降阶映射，对此我们讨论了多种方案。该通用框架能够囊括并推广多种现有MOR技术，例如拉格朗日或哈密顿动力学中的结构保持，以及输运主导问题中具有重要意义的非线性投影。这一统一抽象可推导不同方法的共享理论性质（如精确复现结果）。为衔接现有研究成果，我们展示了多种数据驱动力学构建非线性投影的技术均可纳入本框架。