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Data-driven prediction of unsteady flow fields over a circular cylinder using deep learning

Sangseung Lee, Donghyun You

arXiv:1804.06076v3physics.flu-dyn

TL;DR

The paper addresses prediction of future unsteady cylinder-flow fields, including cases at Reynolds numbers not used for training. It compares CNNs and GANs with and without conservation-law losses using sequence-based flow-field datasets, finding favorable prediction of large-scale structures and stronger long-term performance for GANs and the physics-informed CNN. Small-scale vortex production remains difficult, particularly when spanwise information is absent or recursive errors accumulate.

  • Problem

    High-quality deep-learning prediction of unsteady flow fields is difficult because video modeling must represent complex spatial and temporal features without blurriness.

  • Method

    The study trains and compares CNNs and GANs with and without physical loss functions on sequences of simulated cylinder-flow fields, testing Reynolds numbers excluded from training.

  • Results

    GANs and the multi-scale CNN with physical loss functions are more predictive for long-term recursive forecasting, while all four methods predict the immediate future.

  • Takeaways & Limitations

    The networks predict convection and diffusion of large-scale vortical structures, but production of small-scale vortical structures is difficult to account for.

  • Takeaways & Limitations

    Recursive predictions accumulate errors at flow regimes different from training, and missing spanwise information appears to hinder small-scale vortex generation.

Abstract

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Unsteady flow fields over a circular cylinder are trained and predicted using four different deep learning networks: convolutional neural networks with and without consideration of conservation laws, generative adversarial networks with and without consideration of conservation laws. Flow fields at future occasions are predicted based on information of flow fields at previous occasions. Deep learning networks are trained first using flow fields at Reynolds numbers of 100, 200, 300, and 400, while flow fields at Reynolds numbers of 500 and 3000 are predicted using the trained deep learning networks. Physical loss functions are proposed to explicitly impose information of conservation of mass and momentum to deep learning networks. An adversarial training is applied to extract features of flow dynamics in an unsupervised manner. Effects of the proposed physical loss functions, adversarial training, and network sizes on the prediction accuracy are analyzed. Predicted flow fields using deep learning networks are in favorable agreement with flow fields computed by numerical simulations.

1. Introduction

Deep learning is investigated for predicting unsteady flow fields by learning spatial and temporal flow features, motivated by the difficulty of high-quality video prediction. The study compares CNN and GAN approaches, with and without conservation-law losses, to predict cylinder-flow fields at Reynolds numbers excluded from training.

  • Motivation: Observations of animals adapting to fluid environments motivate learning dominant fluid motions and nonlinear correlations to estimate future flow.The paper frames this as a feasibility study for neural-network prediction of unsteady fluid motion.
  • Related work: The paper examines deep-learning methods for flow prediction after earlier neural-network applications to turbulence modeling and steady bluff-body flow.The cited prior work includes shallow and deep networks, as well as CNN-based steady-flow prediction.
  • Motivation: Video-style deep learning predicts future flow frames from previous frames by learning spatial and temporal features.Such prediction has been difficult because complex spatial and temporal features can produce blurry outputs.
  • Study objective: The study predicts unsteady vortex shedding over a circular cylinder using CNNs and GANs, each with or without conservation-law consideration.Conservation laws are incorporated through loss functions.
  • Study objective: The networks are tested on Reynolds numbers not used during learning, unlike ROMs focused on low-dimensional representations at specified Reynolds numbers.This defines the study’s generalization target rather than representation discovery at a fixed parameter value.

2. Construction of flow field datasets

The dataset combines numerical cylinder-flow simulations across multiple Reynolds numbers and flow regimes with standardized spatial and temporal sampling. Training uses local sequences of five flow fields, while testing uses full interpolated fields at Reynolds numbers excluded from training.

  • Simulation cases: Simulations cover Re_D = 150, 300, 400, 500, 1000, 3000, and 3900 using incompressible Navier–Stokes equations.These cases span two-dimensional vortex shedding, three-dimensional wake transition, and shear-layer transition regimes.
  • Numerical simulations: The numerical solver uses implicit time integration, Crank–Nicolson treatment, second-order central differences, and a structured H-grid with an O-grid around the cylinder.The computational domain is 50D streamwise by 60D cross-flow, with spanwise size 6D below Re_D 1000 and πD otherwise.
  • Dataset representation: Flow data are sampled every δt = 20∆tU∞/D = 0.1 and interpolated onto 250 × 250 grids with four flow variables over a 7D × 7D square domain.The variables are the three velocity components and pressure.
  • Training data: Training samples contain five consecutive local flow fields on a 0.896D × 0.896D domain with 32 × 32 cells.The first four fields form the input and the fifth is the ground-truth target; 500,000 samples are used.
  • Train–test split: Testing uses interpolated full-domain fields, with Reynolds numbers inside the training dataset excluded from the test dataset.This separation evaluates prediction beyond the Reynolds-number cases used for training.

3. Deep learning methodology

The methodology combines convolutional and fully connected components to learn mappings from consecutive flow-field inputs to future fields. It also incorporates multiscale generation, adversarial discrimination, and explicit mass- and momentum-conservation losses.

  • Network training: Training proceeds through feed forward estimation, loss evaluation, back propagation, gradient-based parameter updates, and repetition until the network is sufficiently updated.The loss measures the difference between estimated and ground-truth output tensors.
  • Layer types: Fully connected layers learn complex input-output mappings, whereas convolution layers use kernels and padded feature maps to extract spatial features with fewer parameters.Fully connected parameter counts grow with the numbers of input and output neurons, while convolution layers are typically used for feature extraction.
  • Network inputs: The networks accept four consecutive sets of flow variables as input to learn correlated physical phenomena across the flow variables.Input fields contain normalized velocity and pressure variables; a parameter study found no significant prediction benefit beyond four consecutive inputs.
  • Multiscale generation: The generator uses interpolated inputs at multiple resolutions and rescaling operations to produce predictions on progressively finer grids.Each generative CNN receives a coarse-resolution input and an upscaled flow field from the next network level.
  • Adversarial training: GAN discriminators compare predicted and ground-truth flow fields after convolutional feature extraction and max pooling, with final fully connected outputs bounded by a sigmoid.The discriminator model contains convolutional and fully connected layers, while intermediate features use ReLU activation.
  • Physical losses: Physical losses impose mass and momentum information by minimizing absolute differences in cell-wise mass and momentum fluxes in the x-y plane.The momentum loss includes convective, pressure-gradient, and shear-stress flux contributions.

4. Results

The networks predict future cylinder-flow fields well in single-step forecasts, but recursive accuracy depends on flow regime, architecture, physics losses, and training-step size. GANs and CNNs with physical or adversarial features generally outperform the CNN without physical losses, while small-scale vortices remain difficult to preserve.

  • Network comparison: Errors are smallest near the trained flow regime and larger at higher-complexity regimes, with steep velocity gradients producing elevated boundary-layer errors.Maximum errors are smallest at ReD = 400 and larger at ReD = 150 and 3900.
  • Network comparison: All four networks predict future flow fields particularly well in single-step forecasts, whereas recursive predictions expose accumulated errors and unphysical structures.Case D develops spurious vortical structures near the front stagnation point after several recursive steps.
  • Network comparison: Physical losses reduce mass- and momentum-conservation errors in CNNs, while the GAN without physical losses achieves the best overall L2 and L∞ performance and favorable conservation.Case C improves Lc and Lmom relative to Case D; Case B is reported as best for minimizing L2 and L∞ errors.
  • Captured and missed flow physics: The networks capture large-scale vortex shedding and velocity-deficit evolution, but recursive forecasts misrepresent or dissipate small-scale vortical structures, especially at ReD = 3900.GAN predictions reproduce large-scale convection and diffusion, while high-wavenumber and small-scale content increasingly deviates with recursive forecasting.
  • Captured and missed flow physics: The absence of spanwise information limits learning of the vortex-stretching mechanism responsible for generating small-scale vortices.This limitation helps explain missing small-scale content and unexpected smaller vortices in recursive predictions.
  • Training interval: Training-step size controls the balance between scales: small intervals preserve small structures better, while a large interval lowers errors by emphasizing energetic large-scale motions.Networks trained with 25δt rapidly lose small-scale structures, yet one 25δt prediction has lower errors than 25 recursive 1δt predictions.

5. Conclusion

Deep learning networks predict unsteady cylinder-flow fields at Reynolds numbers absent from training, capturing large-scale vortical motion at lower computational cost than numerical simulations. Recursive predictions preserve large-scale oscillations but dissipate small-scale structures more rapidly.

  • Unsteady flow fields at Reynolds numbers not included during training were predicted using deep learning techniques.
  • The networks predict convection and diffusion of large-scale vortical structures, whereas production of small-scale vortical structures remains difficult to account for.
  • Prediction time intervals can be two to three orders of magnitude larger than conventional numerical-simulation intervals, depending on the training scheme.
  • Large-scale oscillations behind the cylinder are favorably predicted during recursive GAN prediction at ReD = 3900, while small-scale structures dissipate faster than in ground truth.
  • A single time-step advance requires 0.3 seconds on one NVIDIA Titan Xp GPU.
  • The four networks include GANs and multi-scale CNNs, each with or without physical loss functions for conservation of mass and momentum.

Appendix A. Loss functions

The loss functions combine prediction-error terms with gradient, mass-conservation, momentum-conservation, and adversarial objectives. Conservation losses are computed from cell-surface flux differences using discretized flow variables.

  • The prediction loss minimizes differences between predicted and ground truth flow fields.
  • The gradient difference loss uses second-order central differences to sharpen predicted flow fields by penalizing gradient discrepancies.
  • Flow variables on cell surfaces are calculated as arithmetic means of neighboring-cell values, with i,j denoting grid indices and n_x,n_y grid-cell counts.
  • Mass conservation loss minimizes the total absolute mass-flux difference in each cell of the x-y plane.
  • Momentum conservation loss minimizes differences in convective, pressure-gradient, and shear-stress momentum fluxes within each cell.
  • The adversarial loss trains the generator to make the discriminator classify generated flow fields as class 1.

Appendix B. Error functions

Error functions compare predicted and ground truth non-dimensionalized flow variables, while conservation errors quantify corresponding cellwise conservation residuals. The proposed loss and error forms use the triangle inequality rather than the original conservation-law forms.

  • Error functions compare non-dimensionalized predicted and ground truth values of u, v, w, and p.
  • The conservation error terms use the cellwise quantities defined for conservation of mass and momentum.
  • The proposed conservation losses and errors are formulated with the triangle inequality instead of the original conservation-law forms.
  • Minimizing these physical loss functions satisfies mass and momentum conservation more strictly, while original-form errors are smaller but vary similarly with δt.

C.1. Effects of numbers of layers and feature maps

The study varies generator depth and feature-map count to assess recursive-prediction errors. The largest tested feature-map set generally performs best, and training errors converge without overfitting.

  • Generator-depth effects are evaluated with GM16, GM18, and GM20 using feature-map set N128.
  • Feature-map effects are evaluated with N32, N64, and N128 using generator configuration GM20.
  • N128 generally yields smaller recursive-prediction errors than N32 and N64, except for Lmom at ReD = 150.
  • The selected generator configuration is GM20 with feature-map set N128.
  • L2, L∞, Lc, and Lmom errors converge during training of the multi-scale CNN without physical loss functions, without overfitting.

C.2. Effects of λadv

The GAN’s adversarial-training coefficient is evaluated across multiple Reynolds numbers and error measures. λadv = 0.10 is selected for subsequent analysis because it provides favorable overall errors.

  • Model configurations: The parameter study uses generator configurations GM16, GM18, and GM20.These configurations are listed alongside number sets N32, N64, and N128 in the generator-model study.
  • Adversarial-coefficient study: λadv = 0.10 is selected because it gives small L∞ errors across all tested Reynolds numbers.It also gives the smallest L2, Lc, and Lmom errors at ReD = 3900.

C.3. Effects of λphy

The physical-loss coefficient λphy is varied in a multi-scale CNN while evaluating prediction errors across Reynolds numbers. λphy = 1.00 is selected because it produces relatively small conservation-related errors.

  • Physical-loss study: λphy = 0.00, 0.10, 0.50, and 1.00 are compared using L2, L∞, Lc, and Lmom errors.The CNN is trained at ReD = 300 and 500 and tested at ReD = 150, 400, and 3900.
  • Physical-loss study: λphy = 1.00 is selected because it shows relatively small Lc and Lmom errors at all tested Reynolds numbers.The adversarial coefficient is fixed at λadv = 0, while λl2 and λgdl are fixed at 1.
  • Configuration dependency: The generator-configuration comparison includes GM16, GM18, and GM20.Figure 19 evaluates their L2, L∞, Lc, and Lmom errors as recursive prediction steps increase.

Appendix D. Flow fields predicted by the GAN trained with a small time-step interval

The appendix visualizes GAN predictions at ReD = 3900 using short recursive intervals. It compares input fields, ground truth, and recursively predicted fields across velocity and pressure components.

  • Short-interval predictions: GAN predictions at ReD = 3900 are shown for cross-stream velocity, spanwise velocity, and pressure at 1δt intervals.The corresponding contour plots are provided in figures 24–27.
  • Network-size context: The number-set study compares N32, N64, and N128 in the multi-scale CNN.Figure 20 reports L2, L∞, Lc, and Lmom errors as recursive prediction steps increase.

Appendix E. Flow fields predicted by the GAN trained with a large time-step interval

The appendix examines GAN predictions at ReD = 3900 over long recursive horizons using 25δt intervals. It presents contour comparisons for velocity and pressure fields and contextual error studies.

  • Long-interval predictions: GAN predictions at ReD = 3900 are shown for cross-stream velocity, spanwise velocity, and pressure at 25δt intervals.Figures 28–30 display predictions at 25δt, 50δt, 75δt, 100δt, and 125δt; streamwise velocity is referenced separately.
  • Parameter-study context: The errors are also examined against training iterations, λadv, and λphy in the accompanying parameter studies.Figure 21 evaluates training-iteration effects, while figures 22 and 23 vary the adversarial and physical-loss coefficients.
  • Short-interval reference: Figures 24–27 provide short-interval contour comparisons for streamwise velocity, cross-stream velocity, spanwise velocity, and pressure.These figures use prediction intervals of 1δt and recursively predict later fields.
  • Long-interval predictions: The long-horizon contour figures compare input sets, ground truth fields, and recursively predicted fields.Fields at later times use previously predicted fields as parts of the input.
  • Contour visualization: The long-horizon plots use 14 contour levels for the velocity fields.The cross-stream velocity range is -0.7 to 0.7, while the spanwise velocity range is -0.5 to 0.5.
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