Source-linked AI summary
Deep Neural Networks for Data-Driven Turbulence Models
Andrea D. Beck, David G. Flad, Claus-Dieter Munz
TL;DR
LES needs closure terms for unresolved subgrid effects, whose relation to coarse-grid flow is unknown; this paper trains convolutional DNNs on DNS/LES data to learn those terms. RNN4 reached ≈47% CC versus RNN0’s ≈34% CC.
Problem
LES closure must represent subgrid effects, but their functional relation to coarse-grid flow is unknown and classical models prescribe it a priori.
Source: §1 · p. 2Method
The paper trains convolutional DNNs on DNS and LES data to learn full subgrid closure terms, including effects of numerical discretization, without a priori functional assumptions.
Source: §1 · p. 2, §3.2 · p. 15, §5 · p. 24Results
RNN4 reached ≈47% CC versus RNN0’s ≈34% CC, and deeper RNNs produced higher correlations overall.
Source: §4.1 · pp. 19–20Takeaways & Limitations
An exact LES is feasible when exact closure terms from a previously conducted DNS are used.
Source: §5 · p. 24Directly applying the ANN model terms to Eqn. 2.7 is not feasible because the approximate closure does not permit operator cancellation.
Source: §4.3 · p. 23
Abstract
from arXiv · showhide
In this work, we present a novel data-based approach to turbulence modelling for Large Eddy Simulation (LES) by artificial neural networks. We define the exact closure terms including the discretization operators and generate training data from direct numerical simulations of decaying homogeneous isotropic turbulence. We design and train artificial neural networks based on local convolution filters to predict the underlying unknown non-linear mapping from the coarse grid quantities to the closure terms without a priori assumptions. All investigated networks are able to generalize from the data and learn approximations with a cross correlation of up to 47% and even 73% for the inner elements, leading to the conclusion that the current training success is data-bound. We further show that selecting both the coarse grid primitive variables as well as the coarse grid LES operator as input features significantly improves training results. Finally, we construct a stable and accurate LES model from the learned closure terms. Therefore, we translate the model predictions into a data-adaptive, pointwise eddy viscosity closure and show that the resulting LES scheme performs well compared to current state of the art approaches. This work represents the starting point for further research into data-driven, universal turbulence models.
1. Introduction
The paper develops a data-based DNN approach to learning LES subgrid closures from turbulence data, then derives a practical closure from the learned terms. It frames this as a way to model unresolved effects without prescribing their functional relationship in advance.
1.1. State of the Art
- The derived data-informed eddy-viscosity closure is stable and accurate in LES and performs very well against state-of-the-art LES.The authors present it as a practical model built from the predicted closure terms.Source: §1 · p. 2, §1.1 · p. 3
1. Introduction
- The study learns exact LES subgrid terms by supervised DNN from DNS and LES data of homogeneous isotropic turbulence, without prescribing a closure relationship.The learned reconstruction cancels the coarse-grid LES operator’s influence, so it is not directly suitable for practical LES; the authors fit an explicit closure to the learned perfect model.Source: §1 · p. 2
1.1. State of the Art
- Prior work includes RANS stress learning, LES scale-similarity learning, and neural LES deconvolution.Other cited examples include channel-flow reconstruction and learning subgrid stresses from a single dataset (Gamahara and Hattori).Source: §1.1 · p. 3
- Unlike earlier contributions, this approach learns closure terms directly, uses CNNs to extract structure from volumetric data, constructs a practical model, and separates training, validation, and hidden test data.The separated hidden test data are used to assess generalization.Source: §1.1 · p. 3
- The paper’s strategy proceeds from defining exact closures for a chosen coarse-grid operator and filter to evaluating ANN reconstruction, its inputs, and an eddy-viscosity model.ANN closure predictions are correlated with the exact terms and dissipative, but accumulated time-discretization errors can cause instability when used directly in LES.Source: §1.1 · p. 3
2. Problem Definition: Finding the ”Perfect” LES model
The section defines the perfect LES closure as the target for learning and quality assessment, then describes how DNS data are converted into coarse-grid closure labels. It explains why the closure must account for both subgrid terms and discretization effects.
2.1. The compressible Navier-Stokes equations
- The compressible Navier–Stokes equations govern mass, momentum, and energy conservation for a viscous fluid, with conservative variables comprising density, momentum, and total energy.The equations are closed using the ideal-gas assumption.Source: §2.1 · pp. 3–4
2.2. Perfect LES equations and models
- The perfect LES requires the filtered solution and coarse-grid computation, with a closure that cancels discretization effects while representing the unknown subgrid terms.The operator-filtered formulation is preferred because it includes the discrete operator in the closure; the exact closure is eR(F(U)) − R(F(U)).Source: §2.2 · pp. 5–6
2.3. Perfect LES computations
- DNS data from decaying homogeneous isotropic turbulence were coarsened by a factor of 8 per direction and L2-projected onto P5 to construct LES fields and closure terms.The DNS used a 64^3-element grid with degree-7 polynomials, yielding 512^3 degrees of freedom; the coarse LES grid had 8^3 elements.Source: §2.3 · p. 6
- Preprocessing the DNS-to-LES data is storage-intensive, requiring about 55 TByte for 0.2 T*; the chosen LES operator cancels by design, while the full perfect closure is dissipative.The LES-operator component is slightly antidissipative for the selected dissipative baseline scheme.Source: §2.3 · p. 7
- The perfect LES closely matches filtered DNS, whereas no-model LES builds up high-frequency energy and Smagorinsky LES over-damps near-cutoff modes while retaining excess low-wavenumber energy.The perfect solution recovers the filtered state; imperfect closures alter the extent and shape of large structures and produce more pronounced small-scale artifacts.Source: §2.3 · pp. 7–8, Figure 2 · p. 8, Figure 1 · p. 8
2.4. Generation of Training Data
- Training data came from 20 randomized DNS runs, split into 18 training, one validation, and one hidden test run, sampled from T = 1.0 T* to 2.0 T* at 0.1 T* intervals.This interval balanced increased training data against inter-sample correlation and the cost of additional DNS runs.Source: §2.4 · p. 9
- The network predicts momentum closure terms from coarse velocities and LES operators, and using both input sets improved training compared with using either set alone.Momentum terms were targeted because their closure-to-flux ratio was one to two orders of magnitude larger than for density and energy fluxes; each label preserves a whole LES element’s 3D structure.Source: §2.4 · p. 10
3. Artificial Neural Networks
The section introduces neural networks as trainable nonlinear mappings and motivates convolutional and residual architectures for learning turbulence closures. It describes the chosen residual CNN design and its training choices, including ReLU activations, batch normalization, and a quadrature-weighted squared-error cost.
3. Artificial Neural Networks
- ANNs map inputs to outputs through successive weighted linear operations and nonlinear activations, with training adjusting weights and biases to reduce prediction cost.Training uses forward passes, backpropagation to compute gradients, and gradient-based weight updates; mini-batches and a separate validation set support efficient training and overfitting checks.Source: §3 · pp. 11–12
3.1. Convolutional and Residual Networks
- CNNs retain multidimensional input structure and exploit local geometric relationships, making them suitable for approximating turbulence closure terms.The section also motivates CNNs through their ability to extract hierarchical features from structured data.Source: §3.1 · pp. 12–13
3.1.1. Convolutional Neural Networks
- CNN layers replace dense global connections with local, multidimensional convolution filters whose trainable weights are shared across the input field.Stacked feature maps increase receptive fields in deeper layers and support hierarchical representations, while local connectivity reduces trainable weights and weight sharing enables shift-invariant feature extraction.Source: §3.1.1 · pp. 13–14
3.1.2. Residual Neural Networks
- Residual blocks add a linear skip mapping to convolutional layers so that the learned branch approximates only the nonlinear residual, helping address accuracy degradation with depth.A preactivation design with identity mappings enabled training residual networks exceeding 1000 layers, as reported by He et al. (2016).Source: §3.1.2 · pp. 14–15
3.2. Network Architecture and Hyperparameters
- The study uses residual CNNs for the LES closure; its point-to-point MLP reference achieved less than half the cross-correlation of the cell-to-cell RNN approach.The residual network uses two convolutional-layer stacks per block with identity skip connections, an isotropic filter size of 3, and variable feature-map counts and depth.Source: §3.2 · p. 15
- The selected networks use ReLU activations and batch normalization, with a quadrature-weighted squared-error cost and mini-batch Adam optimization.The quadrature weights counter bias from nonuniform Legendre-Gauss-Lobatto node sampling; optimized activation variants showed no consistent accuracy improvement.Source: §3.2 · p. 16
4. Results
The networks learn meaningful nonlinear approximations of DNS closure terms from coarse-grid data, with accuracy sensitive to input features. A bounded, data-informed eddy-viscosity closure yields a stable LES scheme, whereas applying the approximate ANN terms directly does not provide long-term stability.
4.1. ANN Training Results
- Deeper RNNs generally achieved higher correlations, but gains saturated for d > 4; the deepest RNN showed validation-cost increase after approximately 40,000 iterations, consistent with overfitting and data-limited training.Training and validation costs otherwise continued to drop or flatten without increasing.Source: §4.1 · pp. 18–20
- RNN4 achieved ≈47% CC and RNN0 ≈34% CC on a test sample; inner-element CC exceeded 0.7, while surface correlation was weaker.Both predictions captured general scales, with RNN4 in better agreement; the weaker surface correlation was attributed as likely due to non-isotropy and filter-kernel effects at boundaries.Source: Figure 9 · p. 19, §4.1 · p. 20
4.2. Sensitivity to Input Features
- Using velocities and LES operators together improved learning over either feature set alone, while adding all available coarse-scale quantities gave no improvement and component-wise training was weaker than simultaneous prediction.Although velocities had only weak target correlations, including them improved generalization; weaker component-wise results suggest off-directional coarse-grid components matter.Source: §4.2 · pp. 20–21, Table 5 · p. 21
4.3. Model Construction and Large Eddy Simulation
- Directly applying the approximate ANN closure terms was not practical: despite dissipative predictions, high-frequency errors accumulated and caused long-term instability, even at small CFL numbers.Reducing the timestep improved short-term agreement with filtered LES but did not prevent later stability problems.Source: §4.3 · p. 22, Figure 10 · p. 22
- The bounded pointwise ANN eddy-viscosity closure closely matched filtered DNS in kinetic-energy evolution and spectra, remained stable, and compared favorably with state-of-the-art LES using DG schemes.The unbounded ANN viscosity showed noticeable backscatter; limiting it to [−µ0, 20µ0] produced the reported close agreement.Source: §4.3 · pp. 23–24, Figure 11 · p. 23
5. Conclusion and Outlook
The study learns LES closure terms from DNS using neural networks and converts approximate predictions into a pointwise eddy-viscosity closure. The authors present the results as a starting point for potentially universal turbulence models.
5. Conclusion and Outlook
- The fitted eddy-viscosity closure, limited to [−µ0, 20µ0], produced good results compared with other LES approaches.Because the learned terms are approximate, they are not directly applicable; a least-squares fit translates them into a pointwise eddy viscosity.Source: §5 · p. 24
- The best network achieved about CC ≈45%, increasing to CC ≈73% when the outermost points of the three-dimensional input were excluded; performance appeared data-limited.The authors suggest that increasing the network stencil may further improve performance.Source: §5 · p. 24
- For decaying homogeneous isotropic turbulence at medium Reynolds number (Reλ ≈180), the study modeled full closure terms including effects of the LES numerical method.The authors report that exact LES is feasible when supplied with exact closure terms from a prior DNS, maintaining the filtered solution throughout computation.Source: §5 · p. 24
- Residual convolutional networks predicted closure terms from coarse-scale inputs, and including the closure component depending only on coarse-scale quantities improved learning.Source: §5 · p. 24
- Future work will test flow-type, Reynolds- and Mach-number generalization, explore richer neural-network models, and incorporate existing DNS databases.The authors view these results as a starting point for a potentially universal class of turbulence models.Source: §5 · p. 24