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Polytopic Autoencoders with Smooth Clustering for Reduced-order Modelling of Flows

Jan Heiland, Yongho Kim

TL;DR

A polytopic autoencoder architecture that includes a lightweight nonlinear encoder, a convex combination decoder, and a smooth clustering network is proposed that offers a minimal number of convex coordinates for polytopic linear-parameter varying systems while achieving acceptable reconstruction errors compared to proper orthogonal decomposition.

Abstract

With the advancement of neural networks, there has been a notable increase, both in terms of quantity and variety, in research publications concerning the application of autoencoders to reduced-order models. We propose a polytopic autoencoder architecture that includes a lightweight nonlinear encoder, a convex combination decoder, and a smooth clustering network. Supported by several proofs, the model architecture ensures that all reconstructed states lie within a polytope, accompanied by a metric indicating the quality of the constructed polytopes, referred to as polytope error. Additionally, it offers a minimal number of convex coordinates for polytopic linear-parameter varying systems while achieving acceptable reconstruction errors compared to proper orthogonal decomposition (POD). To validate our proposed model, we conduct simulations involving two flow scenarios with the incompressible Navier-Stokes equation. Numerical results demonstrate the guaranteed properties of the model, low reconstruction errors compared to POD, and the improvement in error using a clustering network.

Polytopic Autoencoders with Smooth Clustering for Reduced-order Modelling of Flows

TL;DR

A polytopic autoencoder architecture that includes a lightweight nonlinear encoder, a convex combination decoder, and a smooth clustering network is proposed that offers a minimal number of convex coordinates for polytopic linear-parameter varying systems while achieving acceptable reconstruction errors compared to proper orthogonal decomposition.

Abstract

With the advancement of neural networks, there has been a notable increase, both in terms of quantity and variety, in research publications concerning the application of autoencoders to reduced-order models. We propose a polytopic autoencoder architecture that includes a lightweight nonlinear encoder, a convex combination decoder, and a smooth clustering network. Supported by several proofs, the model architecture ensures that all reconstructed states lie within a polytope, accompanied by a metric indicating the quality of the constructed polytopes, referred to as polytope error. Additionally, it offers a minimal number of convex coordinates for polytopic linear-parameter varying systems while achieving acceptable reconstruction errors compared to proper orthogonal decomposition (POD). To validate our proposed model, we conduct simulations involving two flow scenarios with the incompressible Navier-Stokes equation. Numerical results demonstrate the guaranteed properties of the model, low reconstruction errors compared to POD, and the improvement in error using a clustering network.
Paper Structure (20 sections, 5 theorems, 68 equations, 18 figures, 2 tables, 1 algorithm)

This paper contains 20 sections, 5 theorems, 68 equations, 18 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation and Basic Ideas
  3. Preliminaries and Notation
  4. State Reconstruction within a Polytope
  5. Input converter
  6. Encoder
  7. Differentiable clustering network
  8. Decoder
  9. Training Strategy for PAEs
  10. Polytope Error and Polytopic LPV Representation
  11. Application in Polytopic LPV Approximations
  12. Simulation Results
  13. Data Acquisition and Performance Measures
  14. Dataset: single cylinder
  15. PAEs: single cylinder
  16. ...and 5 more sections

Key Result

Lemma 3.2

Let $\boldsymbol{\rho}=[\rho_1,\rho_2, \cdots, \rho_r]^\top$ and $\boldsymbol{\alpha}=[\alpha_1,\alpha_2, \cdots, \alpha_k]^\top$ with $\rho_i\geq0$, $i\in\{1,2,\cdots , r\}$, $\sum_{i=1}^r\rho_i=1$, $\alpha_j\geq0$, $j\in\{1,2,\cdots , k\}$ and $\sum_{j=1}^k\alpha_j=1$. Then the entries $\alpha_{ij

Figures (18)

  • Figure 1: Polytopic Autoencoder (PAE): "Linear" corresponds to a linear layer, "Conv" refers to a convolutional layer, "GAP" stands for global average pooling, and "KP" is the Kronecker product.
  • Figure 2: Inverted residual block: an efficient approach for designing deep convolutional layers with fewer parameters compared to standard convolutions. (\ref{['sec:encoder']})
  • Figure 3: When latent variables are divided into three clusters in a low-dimensional space, the clustering labels corresponding to the latent variables (red circles) within each circle with radius $m_i$, $i=1,2,3$, are chosen as target labels. Unselected labels are not used for training PAEs. (\ref{['sec:tr']})
  • Figure 4: Conceptual figure depicting the positions of a state $\mathbf{v}$, it's reconstruction $\tilde{\mathbf{v}}$ in a polytope, and it's best approximation $\mathbf{v}^*$. (\ref{['sec:polyerr']})
  • Figure 5: Reconstruction error across the reduced dimension $r$ averaged for 5 runs for the single cylinder case (\ref{['sec:single-pae']}). The shaded regions mark the statistical uncertainty measured through several training runs and appears to be insignificant and, thus, invisible in the plots for most methods.
  • ...and 13 more figures

Theorems & Definitions (15)

  • Definition 3.1
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Remark 4.1
  • Remark 4.2
  • Definition 4.1: Polytope error and best approximation
  • Lemma 4.3
  • proof
  • Lemma 6.1
  • ...and 5 more