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Convolutional neural network and long short-term memory based reduced order surrogate for minimal turbulent channel flow
Taichi Nakamura, Kai Fukami, Kazuto Hasegawa, Yusuke Nabae, Koji Fukagata
TL;DR
The paper asks whether a machine-learning reduced-order model can represent three-dimensional complex turbulence beyond prior two-dimensional laminar demonstrations. It combines a three-dimensional CNN-AE with an LSTM for latent-space prediction and finds statistical agreement with DNS, while identifying limitations for instantaneous fidelity and more complex flows.
Problem
Previous CNN-AE/LSTM ML-ROM studies were limited to two-dimensional laminar flow fields, leaving applicability to three-dimensional complex flows unresolved.
Method
The ML-ROM combines a three-dimensional CNN-AE that maps flow fields to latent space with an LSTM that predicts latent-vector evolution.
Results
The reproduced turbulent flow fields show good statistical agreement with reference DNS data in the time-ensemble sense, also supported by orbit-based analysis.
Takeaways & Limitations
The CNN-AE/LSTM framework can represent three-dimensional turbulent channel-flow dynamics through temporal evolution in a low-dimensional latent space.
Takeaways & Limitations
Instantaneous ML-ROM fields do not completely match reference fields, and many latent modes may be needed for higher-Reynolds-number turbulence.
Abstract
from arXiv · showhide
We investigate the applicability of machine learning based reduced order model (ML-ROM) to three-dimensional complex flows. As an example, we consider a turbulent channel flow at the friction Reynolds number of $Re_τ=110$ in a minimum domain which can maintain coherent structures of turbulence. Training data set are prepared by direct numerical simulation (DNS). The present ML-ROM is constructed by combining a three-dimensional convolutional neural network autoencoder (CNN-AE) and a long short-term memory (LSTM). The CNN-AE works to map high-dimensional flow fields into a low-dimensional latent space. The LSTM is then utilized to predict a temporal evolution of the latent vectors obtained by the CNN-AE. The combination of CNN-AE and LSTM can represent the spatio-temporal high-dimensional dynamics of flow fields by only integrating the temporal evolution of the low-dimensional latent dynamics. The turbulent flow fields reproduced by the present ML-ROM show statistical agreement with the reference DNS data in time-ensemble sense, which can also be found through an orbit-based analysis. Influences of the population of vortical structures contained in the domain and the time interval used for temporal prediction on the ML- ROM performance are also investigated. The potential and limitation of the present ML-ROM for turbulence analysis are discussed at the end of our presentation.
I. INTRODUCTION
Reduced-order modeling supports analysis and control of complex flows, but the CNN-AE/LSTM ML-ROM had previously been demonstrated only for two-dimensional laminar flows. This paper extends it to three-dimensional turbulent channel flow at Re_τ = 110 and evaluates its applicability.
- Reduced-order modeling helps understand complex flow phenomena and design control laws.
- Intrusive ROM can retain physical information but faces mode-count and numerical-stability limitations for complex flows.
- Autoencoder-based non-intrusive ROM attracts attention because nonlinear activation functions can represent nonlinear low-dimensional mappings.
- Earlier CNN-AE/LSTM ML-ROM studies represented spatio-temporal flow dynamics through low-dimensional latent evolution but were limited to two-dimensional laminar fields.
- The present work extends CNN-AE/LSTM ML-ROM to turbulent channel flow at Re_τ = 110 over a three-dimensional domain.
A. Example: minimal turbulent channel flow at Reτ = 110
The study uses DNS data from a fully developed minimal turbulent channel flow at Re_τ = 110 to train and assess the ML-ROM. The domain and sampling are selected to retain coherent turbulence while remaining computationally manageable.
- The example is a fully developed turbulent channel flow whose DNS training data were verified against spectral DNS results.
- Re_τ = 110 and the minimal domain size (L_x, L_y, L_z) = (πδ, 2δ, 0.5πδ) maintain turbulent coherent structures within available computational resources.
- The simulation uses (N_x, N_y, N_z) = (32, 64, 32) grid points in the selected computational domain.
- The machine-learning attributes are the three velocity components q = {u, v, w}, with 10 000 snapshots sampled at Δt+ = 3.85.
- Seventy percent of snapshots train the model, 30% validate it, and 2 700 additional snapshots provide temporally separated test data.
B. Convolutional neural network based autoencoder
The CNN-AE compresses three-dimensional velocity fields into a nonlinear latent representation and reconstructs them through a decoder. Its multi-scale design uses convolutional paths with different filter sizes to represent structures across flow scales.
- The CNN maps three-dimensional flow-field data through convolutional operations, with max pooling in the encoder and upsampling in the decoder.
- The autoencoder uses the same velocity data q as training input and output, with an encoder for dimension reduction and decoder for dimension extension.
- The latent variable η is treated as representative information for the high-dimensional input when the reconstructed output resembles q.
- Nonlinear autoencoder variables can contain more energetic information than linear POD variables, although orthogonality is not guaranteed.
- The multi-scale CNN-AE uses filter sizes 3, 5, and 7 to account for various scales in complex fluid-flow phenomena.
- The CNN-AE uses the L2 error as its loss function for reconstruction training.
C. Long short-term memory
The LSTM predicts the temporal evolution of CNN-AE latent vectors while retaining sequential history, using a parallel architecture designed for complex time-series data such as turbulence.
- C. Long short-term memory: The LSTM retains previous input information in its cell state to process time-series data.Its cell, input, output, and forget gates support this sequential processing.
- C. Long short-term memory: The LSTM predicts future CNN-AE latent states from the five previous time steps.The previous latent vectors are used as input to predict the next state.
- C. Long short-term memory: Parallel LSTM units are proposed to handle time-series data governed by complex phenomena such as turbulence.The authors report that preliminary tests found the parallel LSTM outperformed the standard LSTM.
- C. Long short-term memory: For testing, the five preceding latent variables are generated from DNS data not used to train the CNN-AE.The preliminary test found that results were not sensitive to the number of input time steps.
D. CNN-LSTM based reduced order model
The ML-ROM encodes consecutive DNS flow fields into latent vectors, recursively predicts their temporal evolution with an LSTM, and reconstructs high-dimensional fields using the CNN decoder.
- D. CNN-LSTM based reduced order model: Five consecutive DNS flow fields are encoded by the CNN-AE and supplied as the initial latent-vector sequence for the LSTM.The LSTM then predicts the latent vector at the next time step.
- D. CNN-LSTM based reduced order model: The LSTM recursively predicts latent-field evolution by feeding each previous output back as the next input.This produces a temporal rollout from the initial encoded sequence.
- D. CNN-LSTM based reduced order model: The trained CNN decoder reconstructs high-dimensional flow fields from the latent fields predicted by the LSTM.This connects low-dimensional temporal prediction to flow-field visualization.
- D. CNN-LSTM based reduced order model: The flow-field reconstructions are visualized using isosurfaces of the second invariant of the velocity gradient tensor at Q+ = 0.01.The figure compares reference DNS with CNN-AE models using large, medium, small, and extra-small latent spaces.
A. Spatial order reduction via CNN
The CNN-AE reduces turbulent channel-flow fields to latent representations, with reconstruction quality depending on latent-space size, training data, vortical content, and wall-normal position.
- Instantaneous reconstruction: Large and medium CNN-AE models reconstruct both large-scale and fine vortex structures, whereas small and extra small models lose small-scale structures through overcompression.The small and extra small latent spaces are insufficient for reconstructing the turbulent channel flow.
- Statistical reconstruction: Large and medium models reasonably match DNS statistics for velocity RMS values and Reynolds shear stress, while stronger compression degrades agreement.The small model remains reasonable for u′+ RMS, whereas the extra small model underestimates all velocity components.
- Spectral reconstruction: The large model shows good agreement with reference DNS in premultiplied streamwise and spanwise energy spectra over the resolved wavenumber range.The spectra assess whether CNN-AE outputs preserve streamwise-velocity energy across scales.
- Training-data dependence: Increasing training snapshots reduces L2 reconstruction error but increases computational cost, creating a trade-off governed by error requirements and available resources.Early stopping prevents overfitting, so computational cost need not grow linearly with snapshot count.
- Vortical-structure dependence: CNN-AE reconstruction error increases with strong vortical content, and the model reconstructs fields better when strong vortex cores are absent.The vortex ratio varies over time, and its correlation with L2 error is examined for all velocity components and individual fluctuations.
- Wall-normal dependence: All models exhibit high error near the wall, likely because low-probability near-wall events are difficult to estimate under L2 minimization.The near-wall observation is consistent with poorer reconstruction when strong vortex cores are present.
B. LSTM based temporal prediction on low-dimensional space
The ML-ROM predicts latent-space temporal dynamics with an LSTM after CNN-AE encoding, reproducing turbulent-channel statistics and orbit behavior under suitable temporal resolution.
- Temporal prediction: The CNN-AE latent vectors are fed into an LSTM to predict high-dimensional flow evolution through low-dimensional dynamics.This forms the reduced-order modeling pipeline.
- Statistical assessment: The parallel-LSTM ML-ROM reproduces velocity statistics and Reynolds shear stress in reasonable agreement with DNS, though center-channel values are overestimated.The standard-LSTM model is the main exception, while recursive input likely contributes to the overestimation.
- Flow structures: The parallel-LSTM reconstruction contains multiple vortex-structure scales also present in DNS, motivating its use in subsequent ML-ROM analyses.The instantaneous fields still need not match DNS exactly.
- Orbit-based assessment: ML-ROM and DNS turbulence-statistics orbits overlap despite distinct time sequences, suggesting similar attractor locations and temporal behavior.The orbit uses TKE, production, and dissipation.
- Time-interval dependence: Four LSTM time intervals are compared because temporal correlation strongly affects supervised turbulence prediction and parameter selection.The baseline M-Short interval is already 100 times the DNS interval.
- Time-interval dependence: Longer LSTM intervals overestimate RMS velocity away from the wall, and the error generally increases with time step.M-Short and M-Medium agree reasonably with DNS, whereas M-Wide and M-Superwide overestimate for y+ ≥20.
- Time-interval dependence: M-Short and M-Medium produce orbit distributions overlapping DNS, whereas M-Superwide overestimates dissipation and departs from the reference orbit.The wide time interval changes the orbit behavior.
IV. CONCLUDING REMARKS
The study extends CNN-AE–LSTM ML-ROM to three-dimensional turbulent channel flow and finds statistical and orbit-based agreement with DNS, while identifying mode-count and temporal-resolution constraints.
- Contribution: A three-dimensional CNN-AE maps turbulent flow fields to latent space, while an LSTM predicts latent-vector evolution for a minimal channel at Reτ = 110.The training data come from DNS.
- Findings: The reconstructed flow fields show good statistical agreement with reference DNS in the time-ensemble sense, and fidelity is also supported by orbit-based analysis.These conclusions concern the turbulent flow representation rather than exact instantaneous matching.
- Parameter dependence: Approximately 1 500 modes are required for the CNN-AE to reconstruct the flow field.The mode requirement is identified through latent-space dependence studies.
- Parameter dependence: A time step that is too wide makes DNS-like reconstruction difficult because temporal correlation is low, so the LSTM interval requires careful selection.This establishes a practical temporal-resolution boundary for the surrogate.
- Limitations: Instantaneous ML-ROM fields do not completely match reference DNS because recursive latent-vector input affects the LSTM prediction.The method can reproduce flow-field behavior without exact instantaneous correspondence.
- Limitations: The large number of latent modes limits application toward more complex flows, including higher-Reynolds-number turbulence.Hierarchical autoencoders and physics-informed losses are suggested as possible future directions.