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HFNO: an interpretable data-driven decomposition strategy for turbulent flows

Marco Cayuela, Vincent Le Chenadec, Peter Schmid, Taraneh Sayadi

Abstract

Fourier Neural Operators (FNOs) have demonstrated exceptional accuracy in mapping functional spaces by leveraging Fourier transforms to establish a connection with underlying physical principles. However, their opaque inner workings often constitute an obstacle to physical interpretability. This work introduces Hierarchical Fourier Neural Operators (HFNOs), a novel FNO-based architecture tailored for reduced-order modeling of turbulent fluid flows, designed to enhance interpretability by explicitly separating fluid behavior across scales. The proposed architecture processes wavenumber bins in parallel, enabling the approximation of dispersion relations and non-linear interactions. Inputs are lifted to a higher-dimensional space, Fourier-transformed, and partitioned into wavenumber bins. Each bin is processed by a Fully Connected Neural Network (FCNN), with outputs subsequently padded, summed, and inverse-transformed back into physical space. A final transformation refines the output in physical space as a correction model, by means of one of the following architectures: Convolutional Neural Network (CNN) and Echo State Network (ESN). We evaluate the proposed model on a series of increasingly complex dynamical systems: first on the one-dimensional Kuramoto-Sivashinsky equation, then on the two-dimensional Kolmogorov flow, and finally on the prediction of wall shear stress in turbulent channel flow, given the near-wall velocity field. In all test cases, the model demonstrates its ability to decompose turbulent flows across various scales, opening up the possibility of increased interpretability and multiscale modeling of such flows.

HFNO: an interpretable data-driven decomposition strategy for turbulent flows

Abstract

Fourier Neural Operators (FNOs) have demonstrated exceptional accuracy in mapping functional spaces by leveraging Fourier transforms to establish a connection with underlying physical principles. However, their opaque inner workings often constitute an obstacle to physical interpretability. This work introduces Hierarchical Fourier Neural Operators (HFNOs), a novel FNO-based architecture tailored for reduced-order modeling of turbulent fluid flows, designed to enhance interpretability by explicitly separating fluid behavior across scales. The proposed architecture processes wavenumber bins in parallel, enabling the approximation of dispersion relations and non-linear interactions. Inputs are lifted to a higher-dimensional space, Fourier-transformed, and partitioned into wavenumber bins. Each bin is processed by a Fully Connected Neural Network (FCNN), with outputs subsequently padded, summed, and inverse-transformed back into physical space. A final transformation refines the output in physical space as a correction model, by means of one of the following architectures: Convolutional Neural Network (CNN) and Echo State Network (ESN). We evaluate the proposed model on a series of increasingly complex dynamical systems: first on the one-dimensional Kuramoto-Sivashinsky equation, then on the two-dimensional Kolmogorov flow, and finally on the prediction of wall shear stress in turbulent channel flow, given the near-wall velocity field. In all test cases, the model demonstrates its ability to decompose turbulent flows across various scales, opening up the possibility of increased interpretability and multiscale modeling of such flows.

Paper Structure

This paper contains 19 sections, 23 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Neural Operator
  3. Hierarchical Fourier Neural Operator
  4. Spectral block
  5. Residual Block
  6. Prediction in physical space
  7. Prediction in wave-number space
  8. Training process
  9. Results
  10. Kuramoto-Sivashinsky equation
  11. Kolmogorov flow
  12. Turbulent channel flow at $Re_{\tau} = 180$
  13. Conclusion
  14. Acknowledgments
  15. Funding Statement
  16. ...and 4 more sections

Figures (11)

  • Figure 1: Generic architecture of HFNO proposed in this work.
  • Figure 2: Three different partitioning strategies are shown from left to right: based on the $\ell_2$-norm, the maximum norm, and a semi-norm depending on a single axis.
  • Figure 3: Architecture of an Echo State Network (ESN)
  • Figure 4: From top to bottom: target flow, prediction, absolute value of the error, contributions of each Fourier layer and residual predictor
  • Figure 5: Kuramoto-Sivashinsky: energy spectrum of the raw data and the prediction of each part of the model
  • ...and 6 more figures