Source-linked AI summary
Predictions of turbulent shear flows using deep neural networks
P. A. Srinivasan, L. Guastoni, H. Azizpour, P. Schlatter, R. Vinuesa
TL;DR
The paper asks whether neural networks can predict temporally evolving turbulent flows accurately enough to capture both statistics and dynamics. It trains and compares MLP and LSTM models on a nine-equation shear-flow model, finding that the LSTM outperforms the MLP while reproducing turbulence statistics, flow structures, and dynamical behavior. The study is exploratory and limited to a low-order near-wall-turbulence representation.
Problem
The paper investigates whether neural networks can predict the temporal dynamics of turbulent flows, with potential relevance to subgrid-scale modeling for more complex simulations.
Method
The study trains and evaluates multilayer perceptron and long short-term memory networks using data from Moehlis et al.'s nine-equation shear-flow model, testing multiple architectures and training choices.
Results
The LSTM outperforms the MLP, yielding lower errors for turbulence statistics and reproducing relevant flow structures and dynamical behavior.
Takeaways & Limitations
The results support using this machine-learning framework as a basis for future data-driven subgrid-scale models of more complex wall-bounded turbulent flows.
Takeaways & Limitations
The study is limited to the Moehlis et al. low-order model with fixed a priori basis functions, while the network predicts only the time coefficients.
Abstract
from arXiv · showhide
In the present work we assess the capabilities of neural networks to predict temporally evolving turbulent flows. In particular, we use the nine-equation shear flow model by Moehlis et al. [New J. Phys. 6, 56 (2004)] to generate training data for two types of neural networks: the multilayer perceptron (MLP) and the long short-term memory (LSTM) network. We tested a number of neural network architectures by varying the number of layers, number of units per layer, dimension of the input, weight initialization and activation functions in order to obtain the best configurations for flow prediction. Due to its ability to exploit the sequential nature of the data, the LSTM network outperformed the MLP. The LSTM led to excellent predictions of turbulence statistics (with relative errors of 0.45% and 2.49% in mean and fluctuating quantities, respectively) and of the dynamical behavior of the system (characterized by Poincaré maps and Lyapunov exponents). This is an exploratory study where we consider a low-order representation of near-wall turbulence. Based on the present results, the proposed machine-learning framework may underpin future applications aimed at developing accurate and efficient data-driven subgrid-scale models for large-eddy simulations of more complex wall-bounded turbulent flows, including channels and developing boundary layers.
I. INTRODUCTION
The study evaluates deep neural networks for predicting the temporal dynamics of a simplified turbulent shear flow. It uses a nine-equation model with Fourier modes representing key near-wall flow structures and their interactions.
- The study assesses whether deep neural networks can predict the temporal dynamics of simple turbulent flows.Satisfactory predictions could support later use of DNNs as subgrid-scale models in large-eddy simulations of more complex cases.
- Training data come from the nine-equation shear-flow model of Moehlis et al. for flow between parallel free-slip walls with sinusoidal body forcing.The model represents a simplified wall-bounded turbulent flow.
- Nine Fourier modes represent the mean profile, streamwise vortices, streaks, instabilities, and their coupling.Instantaneous velocity fields are constructed by superposing these modes.
- Galerkin projection yields nine ordinary differential equations for the time-dependent mode amplitudes, with the model Reynolds number set to Re = 400.U0 and h provide the velocity and length scales, while the Reynolds number determines viscosity and dissipation.
- Only time series that remain turbulent over the full time span are analyzed, excluding trajectories that reach fixed points or periodic orbits.The initial perturbation was randomly generated around the laminar state, and discarded series were identified by their subsequent behavior.
II. PREDICTIONS BASED ON MULTILAYER PERCEPTRONS (MLPS)
This section evaluates MLPs for predicting temporally evolving turbulent-flow amplitudes and statistics. It examines architecture, input-history length, training safeguards, and performance on unseen time series.
- MLP architecture: MLPs map previous amplitude values to the next prediction through layered linear transformations and nonlinear activation functions.The input dimension is d = 9p, where p is the number of previous values used for prediction.
- Training procedure: MLP training uses mean-squared error, validation data, L2 regularization, and early stopping to reduce overfitting.Early stopping halts training when validation loss begins increasing.
- Input-history selection: p = 500 previous values were required for acceptable training losses below 10% when at least two hidden layers were used.This setting gives d = 4,500 input features and m = 9 output quantities.
- Performance: The best tested MLP, MLP4, used 5 hidden layers with 90 neurons per layer and achieved 3.21% mean-profile error but 18.61% streamwise-fluctuation error.Increasing hidden-layer count generally improved amplitude and turbulence-statistics predictions, whereas increasing units did not consistently help.
- Limitations and transition: The relatively large MLP, especially its high-dimensional input, motivated evaluating architectures designed to exploit sequential data more effectively.This limitation provides the rationale for considering recurrent alternatives.
III. PREDICTIONS BASED ON LONG SHORT-TERM MEMORY (LSTM) NETWORKS
LSTM networks outperform MLP baselines in predicting turbulence statistics while reproducing key flow structures and dynamical behavior of the nine-equation model. Increasing training data improves statistical accuracy, although instantaneous mode amplitudes remain sensitive to chaotic divergence.
- LSTM1 and LSTM2 outperform all evaluated MLPs, while LSTM3 performs similarly to the best MLP architecture, MLP4.
- 4.7% and 2.8% are the LSTM1 and LSTM2 deviations in total-shear-stress amplitude, whereas LSTM3 reaches 43.5%.The comparison suggests that the networks can reproduce the relationship between viscous and turbulent stresses.
- 0.83% and 3.44% are the LSTM1 relative errors in mean velocity and fluctuations, compared with 3.21% and 18.61% for MLP4 using 1,000 datasets.
- 0.45% and 2.49% are the LSTM1 relative errors in mean velocity and streamwise fluctuations after training with 10,000 datasets.
- The LSTM1 predictions reproduce streamwise vortices, high- and low-speed streaks, and their spanwise instabilities in reconstructed velocity fields.These structures are characteristic features of the near-wall turbulence cycle represented by the nine-equation model.
- 0.6%, 0.7% and 1.3% are the relative errors for streamwise, wall-normal and spanwise fluctuating vorticities, respectively.
- Despite a validation loss of 5.2 × 10^-9, instantaneous mode-amplitude errors remain high because small perturbations produce different chaotic evolutions.For mode a1, the reported relative error is 13.08%.
- The LSTM reproduces the reference Poincaré-map correlation and has Lyapunov exponents of 0.0264 versus 0.0296 for the reference model.Both trajectories saturate at approximately t ≃ 1,000 after initial divergence.
IV. CONCLUDING REMARKS
The study demonstrates neural-network prediction of temporal dynamics in a low-order turbulent-flow model, with LSTM networks agreeing well with reference turbulence statistics and dynamical behavior. Its scope remains limited to fixed basis functions and predicted time coefficients, while extending the framework to higher-order or more complex flows requires additional basis-function definitions.
- LSTM networks yielded excellent agreement with reference turbulence statistics and dynamical behavior in the nine-equation turbulent-flow model.The study assessed both MLP and LSTM architectures using the Moehlis et al. low-order model.
- The analysis was limited to the Moehlis et al. model, fixed a priori basis functions, and predictions of temporal mode coefficients.The study therefore assessed a low-order representation of near-wall turbulence rather than the full flow field dynamics.
- Extending the approach to higher-order flow representations requires defining suitable basis functions alongside the temporal dynamics.The paper identifies Fourier and proper orthogonal decomposition methods as possible routes for constructing such bases.
- 12 minutes were required to generate 500 LSTM-predicted time series, compared with 6 minutes for integrating the nine-equation model after training.LSTM training with 10,000 time series required around 70 hours on the reported workstation.