A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence

Abstract

In the present study, we investigate different data-driven parameterizations for large eddy simulation of two-dimensional turbulence in the a priori settings. These models utilize resolved flow field variables on the coarser grid to estimate the subgrid-scale stresses. We use data-driven closure models based on localized learning that employs multilayer feedforward artificial neural network (ANN) with point-to-point mapping and neighboring stencil data mapping, and convolutional neural network (CNN) fed by data snapshots of the whole domain. The performance of these data-driven closure models is measured through a probability density function and is compared with the dynamic Smagorinksy model (DSM). The quantitative performance is evaluated using the cross-correlation coefficient between the true and predicted stresses. We analyze different frameworks in terms of the amount of training data, selection of input and output features, their characteristics in modeling with accuracy, and training and deployment computational time. We also demonstrate computational gain that can be achieved using the intelligent eddy viscosity model that learns eddy viscosity computed by the DSM instead of subgrid-scale stresses. We detail the hyperparameters optimization of these models using the grid search algorithm.Keywords Turbulence closure · Deep learning · Neural networks · Subgrid scale modeling · Large eddy simulation 1 Introduction Direct numerical simulation (DNS) of complex fluid flows encountered in many engineering and geophysical applications is computationally unmanageable because of the need to resolve a wide range of spatiotemporal scales. Large eddy simulation (LES) and Reynolds Averaged Navier-Stokes (RANS) modeling are two most commonly used mathematical modeling frameworks that give accurate predictions by considering the interaction between the unresolved and grid-resolved scales. The development of these models is termed as the turbulence closure problem and has been a long-standing challenge in fluid mechanics community [1][2][3][4].In LES, we filter the Navier-Stokes equations using a low-pass filtering operator that separates the motion into small and large scales, and in turn, produces modified equations which are computationally faster to solve than actual Navier-Stokes equations [5][6][7]. The interaction between grid-resolved and unresolved scales is then taken into account by introducing subgrid-scale stress (SGS) term in the modified equation. The main task of the SGS model is to provide mean dissipation that corresponds to the transfer of energy from resolved scales to unresolved scales (the production of energy at large scales is balanced by the dissipation of energy at small scales based on Kolmogorov's theory of turbulence). The dissipation effect of unresolved scales can be included utilizing an eddy viscosity parameterization obtained through grid-resolved quantities. These approaches are called as functional approaches which assume isotropy of small scales to present the average dissipation of t...

Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution

“…In order to gain online computational efficiency at the cost of offline simulations, a physical kernel approach can be imposed in training ML models to enable accurate but fast coarse scale simulation while still intelligently modeling the representation losses and residuals using deep neural networks (i.e., incorporating numerous functional forms of known closure kernels in the training as opposed to static phenomenological models [184,215,216,259]). The training data can be also compressed by using both linear (e.g., matrix factorization [121]) and nonlinear techniques (e.g., variational autoencoders [320], manifold learning [201,207,363] and stochastic neighbor embedding algorithms [140,194,336]), and the causality assessment tools [127] can be employed to explore the correlation between input kernels and output variables in order to remove irrelevant input features from the training.…”

Special issue on machine learning and data-driven methods in fluid dynamics

“…These models would not have been possible without the synergistic combination of both physics and machine learning. In the work of Pawar et al [19], an artificial neural network (multilayer perceptron) and a convolutional neural network are used to construct data-driven subgridscale closure models for two-dimensional turbulence. For the development of the models, they consider a number of spatial points and variables in learning the subgrid-scale stresses at a spatial point.…”

Interpreting neural network models of residual scalar flux

“…However, we do not dwell on other architectures since finding optimal architecture is not the objective of this study. Indeed, non-local approaches to neural network models have been suggested in recent literature (for instance Maulik & San 2017;Duraisamy 2020;Pawar et al 2020). Such non-local neural network models may benefit from exploiting multi-point correlations in resolved flow parameters and may be analogous to non-local mathematical models, such as deconvolutional LES models (see Stolz & Adams (1999), for instance).…”