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Turbulence Scaling from Deep Learning Diffusion Generative Models

Tim Whittaker, Romuald A. Janik, Yaron Oz

TL;DR

This work investigates whether denoising diffusion probabilistic models can learn the statistical structure of 2D turbulent flows by training on vorticity snapshots from DNS and generating new NS solutions. The DDPM employs a U‑Net to model the reverse diffusion process on $256\times256$ vorticity fields, achieving $2.82\times10^{8}$ parameters, and is trained on $5000$ samples downscaled from $512\times512$ simulations with forcing at $k_f\sim40$. Quantitative analyses show the generated fields reproduce the $-5/3$ energy spectrum, structure-function scaling, and local energy-dissipation statistics of the inverse cascade, with novel samples that are not memorized from the training data. The approach offers a data-driven pathway to inflate turbulence statistics and generate realistic flow proxies, though its fidelity depends on the training data and remains bounded by the 2D, moderate-Reynolds-number regime used for training.

Abstract

Complex spatial and temporal structures are inherent characteristics of turbulent fluid flows and comprehending them poses a major challenge. This comprehesion necessitates an understanding of the space of turbulent fluid flow configurations. We employ a diffusion-based generative model to learn the distribution of turbulent vorticity profiles and generate snapshots of turbulent solutions to the incompressible Navier-Stokes equations. We consider the inverse cascade in two spatial dimensions and generate diverse turbulent solutions that differ from those in the training dataset. We analyze the statistical scaling properties of the new turbulent profiles, calculate their structure functions, energy power spectrum, velocity probability distribution function and moments of local energy dissipation. All the learnt scaling exponents are consistent with the expected Kolmogorov scaling. This agreement with established turbulence characteristics provides strong evidence of the model's capability to capture essential features of real-world turbulence.

Turbulence Scaling from Deep Learning Diffusion Generative Models

TL;DR

This work investigates whether denoising diffusion probabilistic models can learn the statistical structure of 2D turbulent flows by training on vorticity snapshots from DNS and generating new NS solutions. The DDPM employs a U‑Net to model the reverse diffusion process on vorticity fields, achieving parameters, and is trained on samples downscaled from simulations with forcing at . Quantitative analyses show the generated fields reproduce the energy spectrum, structure-function scaling, and local energy-dissipation statistics of the inverse cascade, with novel samples that are not memorized from the training data. The approach offers a data-driven pathway to inflate turbulence statistics and generate realistic flow proxies, though its fidelity depends on the training data and remains bounded by the 2D, moderate-Reynolds-number regime used for training.

Abstract

Complex spatial and temporal structures are inherent characteristics of turbulent fluid flows and comprehending them poses a major challenge. This comprehesion necessitates an understanding of the space of turbulent fluid flow configurations. We employ a diffusion-based generative model to learn the distribution of turbulent vorticity profiles and generate snapshots of turbulent solutions to the incompressible Navier-Stokes equations. We consider the inverse cascade in two spatial dimensions and generate diverse turbulent solutions that differ from those in the training dataset. We analyze the statistical scaling properties of the new turbulent profiles, calculate their structure functions, energy power spectrum, velocity probability distribution function and moments of local energy dissipation. All the learnt scaling exponents are consistent with the expected Kolmogorov scaling. This agreement with established turbulence characteristics provides strong evidence of the model's capability to capture essential features of real-world turbulence.
Paper Structure (16 sections, 19 equations, 8 figures)

This paper contains 16 sections, 19 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. 2D Fluid Turbulence
  4. Diffusion Generative Models
  5. Methodology
  6. Model and Architecture
  7. Training Datasets
  8. Training Procedure
  9. Results
  10. Generated Samples
  11. A Test for Memorization
  12. Inverse Cascade
  13. Structure Function
  14. Local Energy Dissipation
  15. Discussion
  16. ...and 1 more sections

Figures (8)

  • Figure 1: Forward noising and backward denoising Markov chains.
  • Figure 2: Left: Evolution in time of the fluid energy, highlighting the attainment of steady state. Right: Energy power spectrum showcasing the $-5/3$ scaling with standard deviation in the shaded regions, indicative of turbulence state in the inertial range.
  • Figure 3: Sample images from the training set (left), and generated by the diffusion model (right).
  • Figure 4: The histogram of cosine distances between 16 generated samples and the 5000 training images (left) and a pair of the most similar sample and training image (right).
  • Figure 5: (a) Energy cascade from the numerical simulation and the DDPM, emphasizing the $-5/3$ slope in the inertial range with a noted discrepancy at lower wavenumbers. The standard deviation is shown in the shaded regions. The vertical line indicates the forcing scale. (b) Measured slope error with standard errors of the fits across varying set sizes, revealing the pronounced influence of the selected range. (c) Measured slope errors using bootstraping.
  • ...and 3 more figures