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Convolutional Autoencoders, Clustering and POD for Low-dimensional Parametrization of Navier-Stokes Equations

Yongho Kim, Jan Heiland

TL;DR

A convolutional autoencoder (CAE) consisting of a nonlinear encoder and an affine linear decoder and consider combinations with k-means clustering for improved encoding performance is proposed.

Abstract

Simulations of large-scale dynamical systems require expensive computations. Low-dimensional parametrization of high-dimensional states such as Proper Orthogonal Decomposition (POD) can be a solution to lessen the burdens by providing a certain compromise between accuracy and model complexity. However, for really low-dimensional parametrizations (for example for controller design) linear methods like the POD come to their natural limits so that nonlinear approaches will be the methods of choice. In this work we propose a convolutional autoencoder (CAE) consisting of a nonlinear encoder and an affine linear decoder and consider combinations with k-means clustering for improved encoding performance. The proposed set of methods is compared to the standard POD approach in two cylinder-wake scenarios modeled by the incompressible Navier-Stokes equations.

Convolutional Autoencoders, Clustering and POD for Low-dimensional Parametrization of Navier-Stokes Equations

TL;DR

A convolutional autoencoder (CAE) consisting of a nonlinear encoder and an affine linear decoder and consider combinations with k-means clustering for improved encoding performance is proposed.

Abstract

Simulations of large-scale dynamical systems require expensive computations. Low-dimensional parametrization of high-dimensional states such as Proper Orthogonal Decomposition (POD) can be a solution to lessen the burdens by providing a certain compromise between accuracy and model complexity. However, for really low-dimensional parametrizations (for example for controller design) linear methods like the POD come to their natural limits so that nonlinear approaches will be the methods of choice. In this work we propose a convolutional autoencoder (CAE) consisting of a nonlinear encoder and an affine linear decoder and consider combinations with k-means clustering for improved encoding performance. The proposed set of methods is compared to the standard POD approach in two cylinder-wake scenarios modeled by the incompressible Navier-Stokes equations.
Paper Structure (14 sections, 7 equations, 12 figures, 1 table)

This paper contains 14 sections, 7 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Incompressible Navier-Stokes Equations
  3. Low-dimensional Parametrization
  4. Proper Orthogonal Decomposition (POD)
  5. Convolutional Neural Network (CNN)
  6. Convolutional Autoencoder (CAE)
  7. Clustered POD (cPOD)
  8. Individual CAE (iCAE)
  9. Low-order Parametrization of the State and the Models
  10. Numerical Examples
  11. Single Cylinder
  12. Double Cylinder
  13. Conclusion
  14. Code Availability

Figures (12)

  • Figure 1: Convolutional Autoencoder for the reconstruction of FEM state vectors
  • Figure 2: Two snapshots of developed velocity states: (left) a single cylinder case at Re=40 (right) a double cylinder case at Re=50
  • Figure 3: (a) Averaged reconstruction errors (b) Averaged relative errors for the single cylinder case (\ref{['sec:num-setup-single-cyl']}).
  • Figure 4: Reconstruction error in $[0,10]$ for the single cylinder case (\ref{['sec:num-setup-single-cyl']}).
  • Figure 5: Convection error in $[0,10]$ for the single cylinder case (\ref{['sec:num-setup-single-cyl']}).
  • ...and 7 more figures