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Deep orthogonal decomposition: a continuously adaptive data-driven approach to model order reduction

Nicola Rares Franco, Andrea Manzoni, Paolo Zunino, Jan S. Hesthaven

TL;DR

A novel deep learning technique for dimensionality reduction and reduced order modeling of parameter dependent partial differential equations, which can accommodate both intrusive and nonintrusive techniques, and stands out as a valuable alternative to other nonlinear techniques, such as deep autoencoders.

Abstract

We develop a novel deep learning technique, termed Deep Orthogonal Decomposition (DOD), for dimensionality reduction and reduced order modeling of parameter dependent partial differential equations. The approach consists in the construction of a deep neural network model that approximates the solution manifold through a continuously adaptive local basis. In contrast to global methods, such as Principal Orthogonal Decomposition (POD), the adaptivity allows the DOD to overcome the Kolmogorov barrier, making the approach applicable to a wide spectrum of parametric problems. Furthermore, due to its hybrid linear-nonlinear nature, the DOD can accommodate both intrusive and nonintrusive techniques, providing highly interpretable latent representations and tighter control on error propagation. For this reason, the proposed approach stands out as a valuable alternative to other nonlinear techniques, such as deep autoencoders. The methodology is discussed both theoretically and practically, evaluating its performances on problems featuring nonlinear PDEs, singularities, and parametrized geometries.

Deep orthogonal decomposition: a continuously adaptive data-driven approach to model order reduction

TL;DR

A novel deep learning technique for dimensionality reduction and reduced order modeling of parameter dependent partial differential equations, which can accommodate both intrusive and nonintrusive techniques, and stands out as a valuable alternative to other nonlinear techniques, such as deep autoencoders.

Abstract

We develop a novel deep learning technique, termed Deep Orthogonal Decomposition (DOD), for dimensionality reduction and reduced order modeling of parameter dependent partial differential equations. The approach consists in the construction of a deep neural network model that approximates the solution manifold through a continuously adaptive local basis. In contrast to global methods, such as Principal Orthogonal Decomposition (POD), the adaptivity allows the DOD to overcome the Kolmogorov barrier, making the approach applicable to a wide spectrum of parametric problems. Furthermore, due to its hybrid linear-nonlinear nature, the DOD can accommodate both intrusive and nonintrusive techniques, providing highly interpretable latent representations and tighter control on error propagation. For this reason, the proposed approach stands out as a valuable alternative to other nonlinear techniques, such as deep autoencoders. The methodology is discussed both theoretically and practically, evaluating its performances on problems featuring nonlinear PDEs, singularities, and parametrized geometries.
Paper Structure (20 sections, 7 theorems, 97 equations, 12 figures, 4 tables, 3 algorithms)

This paper contains 20 sections, 7 theorems, 97 equations, 12 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem setup
  3. An instructive example
  4. Existing techniques in the ROM literature
  5. Deep Orthogonal Decomposition for dimensionality reduction
  6. Architecture design
  7. Model training
  8. Quantifying adaptivity
  9. Numerical experiments: dimensionality reduction
  10. Eikonal equation
  11. Stationary Navier-Stokes flow around a parametrized obstacle
  12. Deep Orthogonal Decomposition for reduced order modeling
  13. General considerations
  14. Learning the DOD coefficients
  15. Numerical experiments: model order reduction
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\Theta\subset\mathbb{R}^{p}$ and $\Theta'\subset\mathbb{R}^{p'}$ be two compact sets, equipped, respectively, with two probability distributions, $\mathbb{P}$ and $\mathbb{Q}$. Let be continuous. For each $\boldsymbol{\mu}\in\Theta$, let $\mathscr{S}_{\boldsymbol{\mu}}:=\{\mathbf{u}_{\boldsymbol{\mu},\boldsymbol{\nu}}\}_{\boldsymbol{\nu}\in\Theta'}\subset\mathcal{F}(\Theta\times\Theta')$ be

Figures (12)

  • Figure 1: A non-comprehensive list of dimensionality reduction techniques in reduced order modeling. While linear techniques can be very effective for problems with a fast-decaying Kolmogorov $n$-width, nonlinear techniques are generally better suited in the opposite scenario. Problems "in between" can present mixed properties, such as a general slow decay compensated by a much faster one in the submanifolds.
  • Figure 2: Spatial domain (left) and projection error analysis (right) for the Navier-Stokes case study, Sections \ref{['subsec:example']} and \ref{['subsec:navier-stokes']}. Left: the domain $\Omega_{\boldsymbol{\mu}}$ is obtained by removing an almond-shaped object from the unit square $(0,1)^{2}$. The parameters $\boldsymbol{\mu}=[\theta,x_{0},y_{0}]$ determine the rotation and the position of the obstacle. Right: decay of the projection error for increasingly larger subspaces, highlighting the differences between the solution manifold $\mathscr{S}$ and its $\boldsymbol{\mu}$-slices $\mathscr{S}_{\boldsymbol{\mu}}$. Here, $\boldsymbol{\mu}_{1}=[0,0.5,0.5]$, $\boldsymbol{\mu}_{2}=[\pi/4,0.4,0.6]$ and $\boldsymbol{\mu}_{3}=[\pi/2,0.7,0.3].$
  • Figure 3: Sketch of a DOD architecture.
  • Figure 4: a) spatial domain for the Eikonal equation example, Section \ref{['subsec:eikonal']}; b) distance to boundary map.
  • Figure 5: Comparison between DOD and other dimensionality reduction strategies for the Eikonal equation example, Section \ref{['subsec:eikonal']}.
  • ...and 7 more figures

Theorems & Definitions (21)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 4
  • ...and 11 more