A machine learning framework for LES closure terms

Abstract

In the present work, we explore the capability of artificial neural networks (ANN) to predict the closure terms for large eddy simulations (LES) solely from coarse-scale data. To this end, we derive a consistent framework for LES closure models, with special emphasis laid upon the incorporation of implicit discretization-based filters and numerical approximation errors. We investigate implicit filter types, which are inspired by the solution representation of discontinuous Galerkin and finite volume schemes and mimic the behaviour of the discretization operator, and a global Fourier cutoff filter as a representative of a typical explicit LES filter. Within the perfect LES framework, we compute the exact closure terms for the different LES filter functions from direct numerical simulation results of decaying homogeneous isotropic turbulence. Multiple ANN with a multilayer perceptron (MLP) or a gated recurrent unit (GRU) architecture are trained to predict the computed closure terms solely from coarse-scale input data. For the given application, the GRU architecture clearly outperforms the MLP networks in terms of accuracy, whilst reaching up to 99.9% cross-correlation between the networks' predictions and the exact closure terms for all considered filter functions. The GRU networks are also shown to generalize well across different LES filters and resolutions. The present study can thus be seen as a starting point for the investigation of data-based modeling approaches for LES, which not only include the physical closure terms, but account for the discretization effects in implicitly filtered LES as well.

Structured Deep Kernel Networks for Data-Driven Closure Terms of Turbulent Flows

“…Table 2 summarizes the results for the different datasets and cases: Both the final MSE-loss as well as the cross-correlation is given. The reported crosscorrelations match the ones reported in [7], thus validating their results: Even in our runs, the ANN using the GRU2 setup performed badly on the DG dataset with only a final cross-correlation of 0.8163, which is way worse than the other listed cross-correlations. The SDKN reached at least the same test accuracies, even without any further hyperparameter optimization.…”

“…In the present work, the dataset described in [1,7] (to which we refer for further details on the following configurations) is used as training set for the machine learning algorithms. This dataset is based on high-fidelity DNS of decaying homogeneous isotropic turbulence (DHIT).…”

“…Before training, the input features of the training dataset were normalized to zero mean and unit variance. The ANN architecture from [7] was used as baseline model, which incorporates a GRU layer to leverage the temporal information in the data. Such a GRU layer was also introduced into the SDKN framework, which clearly demonstrates its modularity.…”

“…( 1) was used without any regularization, since the use of a relatively small batch size (128) is likely to provide a sufficient regularization effect. We remark that both architecture and training are hyperparameter-optimized for the ANN setting, as this was the model of choice in [7]. The experiments were run on an Nvidia GTX1070 using PyTorch implementations of the models.…”

Stable a posteriori LES of 2D turbulence using convolutional neural networks: Backscattering analysis and generalization to higher Re via transfer learning

“…In the non-local approach, which often employs variants of convolutional neural networks (CNNs), the SGS term over the entire domain is estimated in terms of the resolved flow in the entire domain to account for potential spatial correlations (e.g., due to coherent structures) and non-homogeneities in the system. For example, Zanna and Bolton [7,120], Beck and colleagues [4,44], Pawar et al [75], and Subel et al [99] have used this approach for ocean circulation, 3D-DHIT, 2D-DHIT, and forced 1D Burgers' turbulence, respectively.…”