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Nonlinear model reduction for transport-dominated problems

Jan S. Hesthaven, Benjamin Peherstorfer, Benjamin Unger

TL;DR

The paper addresses the inadequacy of linear MOR for transport-dominated problems by surveying nonlinear MOR approaches that leverage nonlinear parametrizations, compatible reduced dynamics, and efficient online solvers. It groups methods into transformation-based, online-adaptive, and instantaneous-residual-minimization categories, clarifying the theoretical and computational trade-offs involved. Key contributions include formalizing nonlinear parametrizations, extending Kolmogorov-width concepts to nonlinear manifolds, and detailing practical strategies (e.g., shifted POD, ADEIM, Neural Galerkin) that enable accurate, online-efficient reduced models for wave-like and moving-structure dynamics. The findings underscore significant progress while pointing to open challenges in structure-preserving reductions, stochastic settings, and non-intrusive approaches, with broad implications for design, optimization, and control in complex systems.

Abstract

This article surveys nonlinear model reduction methods that remain effective in regimes where linear reduced-space approximations are intrinsically inefficient, such as transport-dominated problems with wave-like phenomena and moving coherent structures, which are commonly associated with the Kolmogorov barrier. The article organizes nonlinear model reduction techniques around three key elements -- nonlinear parametrizations, reduced dynamics, and online solvers -- and categorizes existing approaches into transformation-based methods, online adaptive techniques, and formulations that combine generic nonlinear parametrizations with instantaneous residual minimization.

Nonlinear model reduction for transport-dominated problems

TL;DR

The paper addresses the inadequacy of linear MOR for transport-dominated problems by surveying nonlinear MOR approaches that leverage nonlinear parametrizations, compatible reduced dynamics, and efficient online solvers. It groups methods into transformation-based, online-adaptive, and instantaneous-residual-minimization categories, clarifying the theoretical and computational trade-offs involved. Key contributions include formalizing nonlinear parametrizations, extending Kolmogorov-width concepts to nonlinear manifolds, and detailing practical strategies (e.g., shifted POD, ADEIM, Neural Galerkin) that enable accurate, online-efficient reduced models for wave-like and moving-structure dynamics. The findings underscore significant progress while pointing to open challenges in structure-preserving reductions, stochastic settings, and non-intrusive approaches, with broad implications for design, optimization, and control in complex systems.

Abstract

This article surveys nonlinear model reduction methods that remain effective in regimes where linear reduced-space approximations are intrinsically inefficient, such as transport-dominated problems with wave-like phenomena and moving coherent structures, which are commonly associated with the Kolmogorov barrier. The article organizes nonlinear model reduction techniques around three key elements -- nonlinear parametrizations, reduced dynamics, and online solvers -- and categorizes existing approaches into transformation-based methods, online adaptive techniques, and formulations that combine generic nonlinear parametrizations with instantaneous residual minimization.
Paper Structure (110 sections, 120 equations, 2 figures)

This paper contains 110 sections, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model reduction
  3. Linear model reduction and its limitation
  4. Nonlinear model reduction
  5. Outline of the paper
  6. Linear model reduction
  7. Full model
  8. Overview of linear model reduction
  9. The three steps of the offline phase of linear model reduction
  10. Offline Step 1: Collecting training data
  11. Offline Step 2: Training a reduced space
  12. Offline step 3: Constructing a reduced model
  13. Online computations, the lifting bottleneck, and empirical interpolation
  14. Online efficiency
  15. Lifting bottleneck
  16. ...and 95 more sections

Figures (2)

  • Figure 2.1: The snapshot matrix corresponding to the heat-equation example has rapidly decaying singular values, indicating that a low-dimensional subspace can capture the snapshot set efficiently. By contrast, the advection equation leads to a substantially slower decay, highlighting the limitations of linear reduced-space approximations for transport-dominated dynamics. (First published in Notices of the American Mathematical Society in 69, Number 5 (2022), published by American Mathematical Society. © 2022 American Mathematical Society.)
  • Figure 7.1: Neural Galerkin schemes evolve the weight vector as prescribed by the Dirac–Frenkel variational principle and can simultaneously adapt sampling points corresponding to a potential informed by the residual. This adaptation concentrates the sample points in regions that dominate the residual and so yield more efficient residual-norm estimates than static sampling.

Theorems & Definitions (15)

  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 6.1
  • ...and 5 more