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Model reduction on manifolds: A differential geometric framework

Patrick Buchfink, Silke Glas, Bernard Haasdonk, Benjamin Unger

TL;DR

This paper develops a differential geometric framework for model-order reduction on smooth manifolds, unifying linear subspace MOR, nonlinear projections, and structure-preserving MOR. It centers on embedding a low-dimensional reduced manifold $\varphi(\check{\mathcal{M}})$ into a high-dimensional full manifold $\mathcal{M}$ and a compatible reduction map $R$, enabling reduced dynamics through MPG and GMG formulations. By selecting appropriate nondegenerate tensor fields, the framework yields structure-preserving MOR variants for Lagrangian (LMG) and Hamiltonian (SMG) systems, guaranteeing energy or Hamiltonian conservation in reduced models. The authors connect data-driven embedding strategies (linear, quadratic, nonlinear compressive, and autoencoders) to the geometric framework and establish an exact reproduction theory, while outlining extensions to broader structure-preserving systems and PDE contexts.

Abstract

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.

Model reduction on manifolds: A differential geometric framework

TL;DR

This paper develops a differential geometric framework for model-order reduction on smooth manifolds, unifying linear subspace MOR, nonlinear projections, and structure-preserving MOR. It centers on embedding a low-dimensional reduced manifold into a high-dimensional full manifold and a compatible reduction map , enabling reduced dynamics through MPG and GMG formulations. By selecting appropriate nondegenerate tensor fields, the framework yields structure-preserving MOR variants for Lagrangian (LMG) and Hamiltonian (SMG) systems, guaranteeing energy or Hamiltonian conservation in reduced models. The authors connect data-driven embedding strategies (linear, quadratic, nonlinear compressive, and autoencoders) to the geometric framework and establish an exact reproduction theory, while outlining extensions to broader structure-preserving systems and PDE contexts.

Abstract

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.
Paper Structure (51 sections, 15 theorems, 143 equations, 3 figures, 3 tables)

This paper contains 51 sections, 15 theorems, 143 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main contributions
  3. State-of-the-art
  4. MOR and manifolds
  5. Nonlinear mappings and transport MOR
  6. Structure-preserving MOR for Lagrangian and Hamiltonian systems
  7. Structure of the paper
  8. Notation
  9. A primer on differential geometry
  10. Chart and smooth manifold
  11. Diffeomorphism and partial derivative
  12. Tangent and tangent space
  13. Differential and chain rule
  14. Tangent bundle and vector field
  15. Curve and initial value problem
  16. ...and 36 more sections

Key Result

Lemma 2.1

Consider smooth manifolds $\check{\mathcal{M}}, \mathcal{M}$ and smooth mappings $\varphi \in \mathcal{C}^{\infty}(\check{\mathcal{M}}, \mathcal{M})$ and $\varrho \in \mathcal{C}^{\infty}(\mathcal{M}, \check{\mathcal{M}})$ with $\varrho \circ \varphi \equiv \mathop{\mathrm{id}}\nolimits_{\check{\mat

Figures (3)

  • Figure 1: Two intersecting chart domains $U,V \subseteq \mathcal{M}$ with respective chart mappings $x, y$ and transition mappings $x \circ y^{-1}, y \circ x^{-1}$ on $x\left(U\cap V\right)\subseteq\mathbb{R}^{N}$ and $y\left(U\cap V\right)\subseteq\mathbb{R}^{N}$.
  • Figure 2: Schematic illustration of the full manifold $\mathcal{M}$ (dark blue), the set of all solutions $S$ (black), and the approximating embedded submanifold $\varphi(\check{\mathcal{M}})$ (red). The set of solutions is schematically depicted as three separate trajectories that may occur due to a possible discontinuous behavior in the parameter $\mu$.
  • Figure 3: Schematic illustration of the relation between the embedding $\varphi$ and the reduction map $R\left(m, v\right)= \left(\varrho\left(m\right),\mleft.\nulldelimiterspace{R}\mright|_{m}\left(v\right)\right)$ with $m\in\mathcal{M}$.

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Remark 3.1: Parameter dependency
  • Definition 3.3: Reduction map
  • Example 3.4: Linear-subspace MOR
  • Definition 3.5: Reduced-order model
  • Theorem 3.6: Exact reproduction of a solution
  • proof
  • Corollary 3.7: Canonical form
  • proof
  • ...and 31 more