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Comparison of data-driven symmetry-preserving closure models for large-eddy simulation

Syver Døving Agdestein, Benjamin Sanderse

TL;DR

This work compares approaches for constructing symmetry-preserving data-driven LES closures, including tensor-basis neural networks and group-convolutional neural networks, alongside unconstrained convolutional networks and finds that symmetry-preserving models produce more physically consistent velocity-gradient statistics.

Abstract

Symmetries are fundamental to both turbulence and differential equations. The large-eddy simulation (LES) equations inherit these symmetries provided the LES closure respects them. Classical LES closures based on eddy viscosity or scale similarity preserve many of the original symmetries by design. Recently, data-driven neural network closures have been applied to LES to improve accuracy, but stability and generalizability remain challenges, as symmetries are not automatically enforced. In this work, we compare approaches for constructing symmetry-preserving data-driven LES closures, including tensor-basis neural networks (TBNNs) and group-convolutional neural networks, alongside unconstrained convolutional networks. All three data-driven closures outperform classical models in both the functional sense (producing the right amount of dissipation) and the structural sense (stress tensor prediction). While unconstrained networks achieve comparable prediction accuracy, symmetry-preserving models produce more physically consistent velocity-gradient statistics, suggesting that enforcing symmetries improves the quality of the learned closure beyond what aggregate error metrics such as relative tensor prediction errors capture.

Comparison of data-driven symmetry-preserving closure models for large-eddy simulation

TL;DR

This work compares approaches for constructing symmetry-preserving data-driven LES closures, including tensor-basis neural networks and group-convolutional neural networks, alongside unconstrained convolutional networks and finds that symmetry-preserving models produce more physically consistent velocity-gradient statistics.

Abstract

Symmetries are fundamental to both turbulence and differential equations. The large-eddy simulation (LES) equations inherit these symmetries provided the LES closure respects them. Classical LES closures based on eddy viscosity or scale similarity preserve many of the original symmetries by design. Recently, data-driven neural network closures have been applied to LES to improve accuracy, but stability and generalizability remain challenges, as symmetries are not automatically enforced. In this work, we compare approaches for constructing symmetry-preserving data-driven LES closures, including tensor-basis neural networks (TBNNs) and group-convolutional neural networks, alongside unconstrained convolutional networks. All three data-driven closures outperform classical models in both the functional sense (producing the right amount of dissipation) and the structural sense (stress tensor prediction). While unconstrained networks achieve comparable prediction accuracy, symmetry-preserving models produce more physically consistent velocity-gradient statistics, suggesting that enforcing symmetries improves the quality of the learned closure beyond what aggregate error metrics such as relative tensor prediction errors capture.
Paper Structure (27 sections, 66 equations, 11 figures, 3 tables)

This paper contains 27 sections, 66 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Symmetries in large-eddy simulation
  3. Transforms and symmetries
  4. Symmetries of the incompressible Navier-Stokes equations
  5. Symmetries in LES
  6. Tensor-basis neural network
  7. Direct structural closure: Convolutional neural networks
  8. Feed-forward neural network closure
  9. Group-convolutions
  10. Weight sharing for octahedral group convolutions
  11. Results
  12. Velocity fields
  13. Tensor fields
  14. Prediction errors
  15. Total equivariance errors
  16. ...and 12 more sections

Figures (11)

  • Figure 1: Example of weight projection for translation-equivariant group-convolutions in 1D (classical CNNs). For our octahedral group-convolutions, we employ a similar procedure, but without the constraint of locality (enforcing zeros).
  • Figure 2: 2D section of predicted velocity component $u_3$ at $x_3 = 1$ at various times. At the initial time, all LES solutions are equal to the filtered DNS. For Clark, the solution became unstable around $t = 2.1$ and could not be finished.
  • Figure 3: 2D section at $x_3 = 1$ of predicted SFS for a given snapshot $\bar{u}$. The reference SFS $\tau(u)$ is also shown.
  • Figure 4: Relative LES solution errors as a function of time, see \ref{['eq:tensor-error-post']}.
  • Figure 5: Equivariance errors. Left: a-priori errors. Right: a-posteriori errors.
  • ...and 6 more figures

Theorems & Definitions (3)

  • Definition 1: Orientation of space
  • Definition 2: Regular representation
  • Definition 3: Cayley table