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Unsupervised deep learning for super-resolution reconstruction of turbulence

Hyojin Kim, Junhyuk Kim, Sungjin Won, Changghoon Lee

arXiv:2007.15324v1physics.flu-dyn

TL;DR

Super-resolution reconstruction of turbulent flows has typically required paired data, limiting applications where such data are unavailable. The paper applies an unsupervised CycleGAN to unpaired turbulence fields and finds comparable performance to supervised models in paired settings, while reconstructing DNS-quality statistics from unpaired LES data. The authors identify non-unique high-resolution solutions and irregular meshes as remaining issues.

  • Problem

    Supervised turbulence super-resolution requires paired low- and high-resolution data, but practical reconstruction settings may provide only unpaired fields.

  • Method

    The paper applies CycleGAN to learn low- and high-resolution turbulence mappings from unpaired datasets, comparing it with bicubic interpolation, CNN, and cGAN baselines.

  • Results

    CycleGAN performs similarly to the best supervised model in paired examples and reconstructs a high-resolution flow field with DNS-consistent statistics from unpaired LES data.

  • Takeaways & Limitations

    Unsupervised CycleGAN reconstruction is possible without paired turbulence data and may support LES modeling, measurement reconstruction, experiment synchronization, noise removal, and data assimilation.

  • Takeaways & Limitations

    Low-resolution data generally do not uniquely determine high-resolution fields, and the current convolution operation is inappropriate for irregular meshes.

Abstract

from arXiv · show

Recent attempts to use deep learning for super-resolution reconstruction of turbulent flows have used supervised learning, which requires paired data for training. This limitation hinders more practical applications of super-resolution reconstruction. Therefore, we present an unsupervised learning model that adopts a cycle-consistent generative adversarial network that can be trained with unpaired turbulence data for super-resolution reconstruction. Our model is validated using three examples: (i) recovering the original flow field from filtered data using direct numerical simulation (DNS) of homogeneous isotropic turbulence; (ii) reconstructing full-resolution fields using partially measured data from the DNS of turbulent channel flows; and (iii) generating a DNS-resolution flow field from large eddy simulation (LES) data for turbulent channel flows. In examples (i) and (ii), for which paired data are available for supervised learning, our unsupervised model demonstrates qualitatively and quantitatively similar performance as that of the best supervised-learning model. More importantly, in example (iii), where supervised learning is impossible, our model successfully reconstructs the high-resolution flow field of statistical DNS quality from the LES data. This demonstrates that unsupervised learning of turbulence data is indeed possible, opening a new door for the wide application of super-resolution reconstruction of turbulent fields.

1. Introduction

Deep learning has expanded turbulence research, but super-resolution reconstruction has been limited by its reliance on paired low- and high-resolution data. This paper therefore proposes an unsupervised CycleGAN approach and evaluates it across three turbulence reconstruction settings.

  • Motivation: Turbulence is a chaotic, spatio-temporal, multi-scale nonlinear phenomenon that is costly to measure or simulate at sufficiently high resolution.Resolving the smallest turbulent motions becomes increasingly challenging at high Reynolds numbers, particularly for DNS.
  • Background: Earlier neural-network applications to turbulence used shallow learning and were restricted to simple correlations between nearby turbulence quantities.Deep neural networks later broadened turbulence applications as learning algorithms, computational equipment, datasets, and open-source software improved.
  • Research gap: Supervised super-resolution reconstruction requires paired data, motivating a generalized model that can learn from unpaired turbulent datasets.Prior unsupervised learning of turbulent inflow fields suggested that deep networks could learn hidden similarities in unpaired turbulence.
  • Contribution: The proposed unsupervised model applies CycleGAN to reconstruct high-resolution turbulence from unpaired data.The study compares it with bicubic interpolation, CNN, and cGAN models across filtered DNS, partially measured DNS, and independently obtained LES–DNS data.

2. Methodology

The methodology uses CycleGAN to learn low- to high-resolution turbulence mappings without paired samples, while cycle consistency preserves correspondence with the input field. WGAN-GP stabilization, convolutional architectures, and supervised CNN/cGAN baselines support the comparisons where paired data exist.

  • GAN framework: A GAN uses a generator to produce data resembling a target distribution and a discriminator to distinguish generated data from real data.For super-resolution, low-resolution flow fields replace random noise as the generator input.
  • CycleGAN: CycleGAN uses generators G and F to map between unpaired low- and high-resolution datasets X and Y, with discriminators DY and DX evaluating generated fields.G maps X → Y, while F maps Y → X.
  • CycleGAN: Cycle consistency penalizes disagreement between F(G(x)) and x or G(F(y)) and y, helping generated fields remain dependent on their inputs.The loss combines forward and backward cycle-consistency terms based on mean-squared errors.
  • Optimization: The CycleGAN objective combines two GAN losses with a weighted cycle loss, using λ = 10 while generators minimize and discriminators maximize the objective.WGAN-GP replaces the standard GAN loss to stabilize and improve training performance.
  • Architecture: CNN generators and discriminators use convolutional operations, with up-sampling in G and down-sampling in F, DX, and DY to change field dimensions.Leaky ReLU nonlinearities and fully connected discriminator layers are also used.
  • Baselines and data: CNN and cGAN baselines train on paired low- and high-resolution fields, whereas CycleGAN intentionally uses shuffled unpaired fields and remains applicable to naturally unpaired LES–DNS data.The cGAN additionally conditions its discriminator on the low-resolution input, so it requires paired data.

3. Results and discussion

Across homogeneous isotropic and wall-bounded turbulence, CycleGAN reconstructed high-resolution fields and statistics from unpaired data with performance similar to the strongest supervised cGAN. It also succeeded for LES-to-DNS reconstruction, where paired supervised training was unavailable.

  • 3.1. Example 1: filtered homogeneous isotropic turbulence: At Reλ = 418, CycleGAN was trained on unpaired filtered-DNS and DNS fields, while CNN and cGAN used paired fields for homogeneous isotropic turbulence.The study simultaneously trained all three velocity components and compared CycleGAN with bicubic interpolation, CNN, and cGAN.
  • 3.1. Example 1: filtered homogeneous isotropic turbulence: CycleGAN reconstructed velocity fields similarly to DNS and cGAN across r = 4, 8, and 16, although pointwise differences increased at r = 16.CNN had the lowest MSE, but cGAN and CycleGAN better reconstructed small-scale structures; the authors caution that MSE may not fully measure turbulence super-resolution quality.
  • 3.1. Example 1: filtered homogeneous isotropic turbulence: CycleGAN and cGAN recovered DNS-like vorticity structures, vorticity probability distributions, and high-wavenumber energy, unlike bicubic interpolation and CNN.CycleGAN’s vorticity performance deteriorated somewhat at r = 16, while its probability density and transverse energy spectra recovered DNS characteristics across the tested ratios.
  • 3. Results and discussion: Overall, CycleGAN provided statistically accurate high-resolution fields from unpaired turbulence data and performed nearly as well as cGAN where supervised comparison was possible.The results support CycleGAN as an effective model across filtered, partially measured, and LES-based reconstruction tasks.
  • 3.2. Example 2: measured wall-bounded turbulence: For sparse pointwise measurements in turbulent channel flow, CycleGAN reconstructed streaks, wall-normal small structures, and DNS-like energy spectra similarly to cGAN.The comparison represents an experimental-style setting in which low-resolution measurements are not local averages but values sampled on sparse grids.
  • 3.2. Example 2: measured wall-bounded turbulence: Adding pixel loss reduced errors at measured positions and across the reconstructed field, while also eliminating phase errors within the resolved wavenumber range.The phase accuracy applied to all velocity components where κ ⩽κ_cutoff, while small-scale structures were reconstructed beyond the measured resolution.
  • 3.3. Example 3: application to large-eddy simulation data: With unpaired LES and DNS data, CycleGAN alone reconstructed DNS-like small-scale velocity features while preserving LES large-scale structure.Bicubic interpolation and CNN failed to generate DNS small scales, and cGAN produced fields whose DNS fidelity was difficult to establish in this setting.
  • 3.3. Example 3: application to large-eddy simulation data: CycleGAN accurately reconstructed differentiated high-order components and closely reproduced DNS probability distributions and statistics from LES inputs.Its velocity and vorticity distributions matched DNS except for the low-speed range of streamwise velocity, whereas supervised models showed larger discrepancies in several statistics.

4. Conclusion

The study shows that CycleGAN can reconstruct small-scale turbulence structures from unpaired low- and high-resolution data. It performs similarly to supervised models when paired data exist and reconstructs DNS-quality statistics from unpaired LES data, while several practical limitations remain.

  • CycleGAN reconstructs small-scale turbulence structures when low- and high-resolution fields are unpaired.The study compares it with bicubic interpolation and supervised CNN and cGAN models.
  • CycleGAN reproduces energy spectra and vorticity probability-density statistics well despite using unpaired data in homogeneous isotropic turbulence.Bicubic interpolation and CNN did not adequately recover small-scale structures, including at large resolution ratios.
  • CycleGAN reconstructs channel-flow fields from partially measured DNS data with flow fields and statistics similar to DNS and performance comparable to cGAN.Pixel loss was introduced to reduce phase shifts from pointwise measurements.
  • CycleGAN reconstructs DNS-quality flow fields from LES data when paired DNS data are unavailable, whereas supervised CNN and cGAN models fail to produce small scales.The conclusion reports agreement across velocity and vorticity distributions, energy spectra, and two-point statistics.
  • The method remains limited by non-unique high-resolution solutions, irregular-mesh inputs, and the assumption of sufficient high-resolution training data.The authors suggest future use of graph CNNs, data augmentation, and physics-informed neural networks.
  • The authors expect CycleGAN to assist LES modeling and high-resolution reconstruction of measurement data such as PIV.Proposed applications include producing paired data for subgrid-scale-model development and synchronizing model evaluations.

Appendix A. Network architecture and hyperparameters of deep learning model

For resolution ratio eight, CycleGAN uses two generators and two discriminators with complementary convolutional, up-sampling, and down-sampling operations. The comparison CNN and cGAN models share the principal generator architecture and most hyperparameters.

  • CycleGAN contains two generators, G and F, and two discriminators, DX and DY.G reconstructs the high-resolution turbulent field; F, DX, and DY use convolution and down-sampling operations.
  • Generator G repeatedly applies convolution and up-sampling operations to reconstruct the high-resolution turbulent field.
  • CNN and cGAN use the same generator network as CycleGAN’s G for supervised comparison.The cGAN discriminator is nearly the same as CycleGAN’s DY, differing in input channel size.
  • CNN, cGAN, and CycleGAN use the same hyperparameters except for CNN’s learning rate and total iterations.CNN starts at learning rate 0.0005 and reduces it by one-fifth when validation error stops decreasing.

Appendix B. Test in the outer-region of wall-bounded turbulent flows

In outer-region channel-flow reconstruction from partially measured data, CycleGAN matches cGAN in field accuracy and closely reproduces DNS spectra, while bicubic interpolation and CNN capture mainly large-scale structures.

  • CycleGAN reconstructs velocity fields from partially measured data at y+ = 15 and 100 using additional point-by-point pixel loss.
  • At y+ = 100, CycleGAN reconstructs velocity fields as accurately as cGAN, the best-performing supervised model.
  • Bicubic interpolation and CNN capture only large-scale structures compared with DNS.
  • CycleGAN’s one-dimensional energy spectra are similar to DNS and cGAN, with slight errors at a few wave numbers.The reported error is related to the up-sampling scheme in generator G and could be avoided by changing the interpolation method.

Appendix C. Validation of large eddy simulation

The LES validation uses turbulent channel-flow simulations at Reτ = 1,000 with two subgrid-scale models and compares their mean and RMS velocity profiles and reconstructed spectra.

  • LES channel flow uses periodic streamwise and spanwise boundaries, a constant mean pressure gradient, and no-slip top and bottom walls.The governing equations are filtered incompressible Navier–Stokes equations.
  • Reconstructed wall-normal velocity spectra at y+ = 15 compare CycleGAN models trained with Vreman- and Smagorinsky-model LES data.
  • The validation examines mean velocity and RMS velocity profiles from the Vreman and Smagorinsky LES models.
  • The simulations use Smagorinsky and Vreman subgrid-scale models with Cs = 0.17 for both models.Van Driest damping is additionally applied to the Smagorinsky model.
  • Both LES cases use a 128 × 256 × 128 grid and domain size 2πδ × 2δ × πδ at Reτ = 1,000.The resolution ratio relative to DNS is four in both streamwise and spanwise directions.
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