Source-linked AI summary
Deep Neural Networks for Data-Driven Turbulence Models
Andrea D. Beck, David G. Flad, Claus-Dieter Munz
TL;DR
The paper addresses data-based closure modeling for LES when the relationship between coarse-grid inputs and unresolved effects is unknown. It learns closure terms from DNS and LES data using neural networks, showing that exact closures can make LES feasible while learned approximations achieve measurable correlation with the targets.
Problem
LES requires closure terms for unresolved subgrid effects, whose functional relationship with grid-resolved quantities is not assumed in this data-based formulation.
Method
The authors use supervised deep neural networks with local convolution filters to learn exact LES subgrid closures, including effects from the underlying discretization operators.
Results
Exact closure terms from DNS make exact LES feasible, while trained networks achieve cross-correlation up to 47% overall and approximately 73% for inner elements.
Takeaways & Limitations
The study supports data-based LES closure modeling for decaying homogeneous isotropic turbulence and provides a starting point for further data-driven turbulence-model research.
Takeaways & Limitations
The study uses decaying homogeneous isotropic turbulence at medium Reynolds number and requires expensive DNS storage, approximately 55 TByte for one described preparation interval.
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
Deep neural networks are presented as data-driven approximators for unknown relationships in fluid mechanics, particularly where first-principles equations require closure terms. The paper applies this approach to LES turbulence modelling using supervised learning from DNS and LES data.
- DNNs are useful when input-output structure is unknown but sufficient training data and computing resources are available.
- Machine-learning closures can augment governing fluid-mechanics equations where unresolved physical effects or scales require additional information.
- The work learns LES subgrid terms from DNS and LES data for homogeneous isotropic turbulence without imposing an a priori functional relationship.
- Unlike classical explicit models, the resulting closure is a data-based approximation of an existing but unknown functional relationship.
- Prior neural-network turbulence studies include learning Reynolds stresses for RANS, reconstructing channel-flow fields, and early LES-modelling applications.
2. Problem Definition: Finding the ”Perfect” LES model
The paper defines a perfect LES by requiring the coarse-grid equation to recover the filtered DNS solution while accounting explicitly for subgrid and discretization effects. It then constructs exact closure data from DHIT simulations as targets for data-based neural-network models.
- Perfect LES definition: A perfect LES must recover the filtered solution U while computing every term on the coarse grid.The coarse-grid formulation is designed so the exact closure restores the filtered DNS evolution.
- Perfect LES definition: The perfect closure cancels discretization effects, whereas imperfect models generally leave residual discretization errors and produce solutions different from U.The cancellation depends on retaining the exact right-hand side of the perfect LES equation.
- Perfect LES definition: The exact closure is the difference between the coarse-grid discretized operator and the filtered DNS divergence operator.This closure term is used as training data for a data-based discrete model.
- Data generation: The study generates closure data from decaying homogeneous isotropic turbulence using DNS and a DGSEM discretization on a periodic cube.The selected test case has Reynolds number based on the Taylor microscale of about 180.
- Perfect LES computations: In velocity-field comparisons, perfect-model LES closely matches filtered DNS, while no-model and Smagorinsky LES alter small-scale structures and spectral energy.No-model LES lacks sufficient dissipation, whereas Smagorinsky closure over-damps near-cutoff modes and leaves low wave numbers too energetic.
- Data-based closure model: The neural-network inputs preserve three-dimensional element structure and combine coarse-scale velocities with LES operators applied to the coarse field.This layout motivates convolutional and recurrent neural-network architectures.
3. Artificial Neural Networks
Artificial neural networks map inputs to outputs through successive linear and nonlinear operations, while CNNs preserve spatial structure through local connectivity and convolution. The paper focuses on CNN-type residual architectures because they are suited to structural data and outperform the reference point-to-point MLP.
- Neural-network fundamentals: Artificial neural networks compose weighted linear operations and nonlinear activations to map an input vector X to an output vector Y.Each layer combines activations from the previous layer with weights and a bias before applying an activation function.
- Convolutional neural networks: CNNs retain multidimensional sample structure and use local receptive fields, convolutional filters, and shared weights to represent spatial relationships.Stacking convolutional layers expands deeper neurons’ effective receptive fields and supports hierarchical feature representations.
- Neural-network fundamentals: MLPs use dense connections between adjacent layers, giving each neuron a receptive field spanning all neurons in the previous layer.Network depth and neurons per layer influence model capability and training requirements.
- Convolutional neural networks: CNNs reduce trainable parameters relative to MLPs, making training more efficient and robust while enabling deeper networks.The same filter is applied across a feature map through weight sharing.
- Network choice for LES closure: The study focuses on RNN architectures implemented with CNN-type networks and uses the preserved three-dimensional structure of LES closure targets.The reference point-to-point MLP achieved less than half the cross-correlation obtained with the RNN approach.
- Network choice for LES closure: The selected networks use ReLU activations and batch normalization within residual blocks.Batch normalization normalizes layer inputs within each training batch to improve learning speed and robustness.
4. Results
The networks learned meaningful nonlinear approximations of LES closure terms, with accuracy depending on depth, feature choice, and available training data. Direct ANN closures were not long-term stable, but a data-adaptive eddy-viscosity formulation produced a stable and accurate LES scheme.
- ANN Training Results: All investigated networks learned from the training data, with validation and training costs generally decreasing or flattening without evident overfitting.The largest RNN showed a validation-cost increase after approximately 40,000 iterations.
- ANN Training Results: Over 0.7 cross-correlation was learned for inner elements, while surface correlations were significantly weaker because of boundary non-isotropy and filter effects.The authors attribute the boundary weakness to non-isotropy and the filter kernel at element boundaries.
- ANN Training Results: The networks generated nonlinear feature combinations rather than merely reproducing a linear input mapping, and deeper RNNs generally achieved higher prediction correlations.Correlation gains saturated for depths above four because of overfitting and limited training data.
- Sensitivity to Input Features: Including both coarse-grid velocities and LES operators improved generalization, whereas using either feature group alone produced considerably lower training accuracy.The LES operators correlate more strongly with the targets, while velocities still improve generalization despite weak direct correlations.
- Model Construction and Large Eddy Simulation: Direct ANN closure terms were dissipative initially but lacked long-term stability as high-frequency errors accumulated, even when smaller CFL numbers improved short-term agreement.The direct closure was therefore judged impractical for long-term LES evolution.
- Model Construction and Large Eddy Simulation: The learned terms were instead converted into a data-informed adaptive eddy-viscosity closure, yielding a stable and accurate scheme that compared favorably with a current state-of-the-art LES approach.Limiting the ANN viscosity to [−µ0, 20µ0] addressed noticeable backscatter and produced close agreement with filtered DNS.
5. Conclusion and Outlook
The work investigates a data-based LES turbulence-modelling paradigm using full closure terms that include effects from the underlying numerical method. It demonstrates feasibility for decaying homogeneous isotropic turbulence and establishes a basis for further data-driven turbulence-model research.
- The study investigates a new data-based paradigm for turbulence modelling in Large Eddy Simulation.
- The canonical model problem is decaying homogeneous isotropic turbulence at medium Reynolds number, Reλ ≈180.
- The learned closure targets full subgrid terms, including effects introduced by the numerical method used for LES.
- Exact LES is feasible when exact closure terms obtained from a previously conducted direct numerical simulation are used.