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Interpreting neural network models of residual scalar flux

Published online by Cambridge University Press:  23 November 2020

G. D. Portwood Open the ORCID record for G. D. Portwood [Opens in a new window] ,
B. T. Nadiga ,
J. A. Saenz  and
D. Livescu Open the ORCID record for D. Livescu [Opens in a new window]
G. D. Portwood*
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
B. T. Nadiga
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
J. A. Saenz
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
D. Livescu
Affiliation:
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
*
†Email address for correspondence: portwood@lanl.gov
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Abstract

We show that, in addition to providing effective and competitive closures, when analysed in terms of the dynamics and physically relevant diagnostics, artificial neural networks (ANNs) can be both interpretable and provide useful insights into the on-going task of developing and improving turbulence closures. In the context of large-eddy simulations (LES) of a passive scalar in homogeneous isotropic turbulence, exact subfilter fluxes obtained by filtering direct numerical simulations are used both to train deep ANN models as a function of filtered variables, and to optimise the coefficients of a turbulent Prandtl number LES closure. A priori analysis of the subfilter scalar variance transfer rate demonstrates that learnt ANN models outperform optimised turbulent Prandtl number closures and Clark-type gradient models. Next, a posteriori solutions are obtained with each model over several integral time scales. These experiments reveal, with single- and multi-point diagnostics, that ANN models temporally track exact resolved scalar variance with greater accuracy compared to other subfilter flux models for a given filter length scale. Finally, we interpret the artificial neural networks statistically with differential sensitivity analysis to show that the ANN models feature a dynamics reminiscent of so-called ‘mixed models’, where mixed models are understood as comprising both a structural and functional component. Besides enabling enhanced-accuracy LES of passive scalars henceforth, we anticipate this work to contribute to utilising neural network models as a tool in interpretability, robustness and model discovery.

JFM classification

Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulence simulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 907 , 25 January 2021 , A23
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Balarac, G., Le Sommer, J., Meunier, X. & Vollant, A. 2013 A dynamic regularized gradient model of the subgrid-scale scalar flux for large eddy simulations. Phys. Fluids 25 (7), 075107.CrossRefGoogle Scholar
Bardina, J., Ferziger, J. H. & Reynolds, W. C. 1980 Improved subgrid-scale models for large-eddy simulation. In 13th Fluid and Plasmadynamics Conference, p. 1357.Google Scholar
Beck, A., Flad, D. & Munz, C.-D. 2019 Deep neural networks for data-driven LES closure models. J. Comput. Phys. 398, 108910.Google Scholar
Brunton, S. L., Noack, B. R. & Koumoutsakos, P. 2020 Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52 (1), 477–508.Google Scholar
Brunton, S. L., Proctor, J. L. & Kutz, J. N. 2016 Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113 (15), 3932–3937.CrossRefGoogle ScholarPubMed
Clark, R. A., Ferziger, J. H. & Reynolds, W. C. 1979 Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J. Fluid Mech. 91, 1–16.Google Scholar
Cybenko, G. 1989 Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2 (4), 303–314.CrossRefGoogle Scholar
Daniel, D., Livescu, D. & Ryu, J. 2018 Reaction analogy based forcing for incompressible scalar turbulence. Phys. Rev. Fluids 3, 094602.CrossRefGoogle Scholar
Duraisamy, K. 2020 Machine learning-augmented Reynolds-averaged and Large Eddy Simulation Models of turbulence. arXiv:2009.10675.Google Scholar
Gamahara, M. & Hattori, Y. 2017 Searching for turbulence models by artificial neural network. Phys. Rev. Fluids 2, 054604.Google Scholar
Gilpin, L. H., Bau, D., Yuan, B. Z., Bajwa, A., Specter, M. & Kagal, L. 2018 Explaining explanations: an overview of interpretability of machine learning. In 2018 IEEE 5th International Conference on Data Science and Advanced Analytics (DSAA), pp. 80–89. IEEE.Google Scholar
Kingma, D. P. & Ba, J. 2014 ADAM: A method for stochastic optimization. arXiv:1412.6980.Google Scholar
Kutz, J. N. 2017 Deep learning in fluid dynamics. J. Fluid Mech. 814, 1–4.Google Scholar
Langford, J. A. & Moser, R. D. 1999 Optimal LES formulations for isotropic turbulence. J. Fluid Mech. 398, 321–346.Google Scholar
Leonard, A. 1974 Energy cascade in LES of turbulent fluid flows. Adv. Geophys. 18A, 237–248.Google Scholar
Ling, J., Kurzawski, A. & Templeton, J. 2016 Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.CrossRefGoogle Scholar
Lu, H. & Porté-Agel, F. 2013 A modulated gradient model for scalar transport in large-eddy simulation of the atmospheric boundary layer. Phys. Fluids 25 (1), 015110.CrossRefGoogle Scholar
Maulik, R. & San, O. 2017 A neural network approach for the blind deconvolution of turbulent flows. J. Fluid Mech. 831, 151–181.CrossRefGoogle Scholar
Maulik, R., San, O., Jacob, J. D. & Crick, C. 2019 a Sub-grid scale model classification and blending through deep learning. J. Fluid Mech. 870, 784–812.CrossRefGoogle Scholar
Maulik, R., San, O., Rasheed, A. & Vedula, P. 2019 b Subgrid modelling for two-dimensional turbulence using neural networks. J. Fluid Mech. 858, 122–144.CrossRefGoogle Scholar
Meneveau, C. 1994 Statistics of turbulence subgrid-scale stresses: necessary conditions and experimental tests. Phys. Fluids 6 (2), 815–833.CrossRefGoogle Scholar
Moghaddam, A. A. & Sadaghiyani, A. 2018 A deep learning framework for turbulence modeling using data assimilation and feature extraction. arXiv:1802.06106.Google Scholar
Nikolaou, Z. M., Chrysostomou, C., Minamoto, Y. & Vervisch, L. 2019 Neural network-based modelling of unresolved stresses in a turbulent reacting flow with mean shear. arXiv:1904.08167.Google Scholar
Overholt, M. R. & Pope, S. B. 1996 Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence. Phys. Fluids 8, 3128–3148.CrossRefGoogle Scholar
Overholt, M. R. & Pope, S. B. 1998 A deterministic forcing scheme for direct numerical simulations of turbulence. Comput. Fluids 27, 11–28.CrossRefGoogle Scholar
Pawar, S., San, O., Rasheed, A. & Vedula, P. 2020 A priori analysis on deep learning of subgrid-scale parameterizations for kraichnan turbulence. Theor. Comput. Fluid Dyn. 34, 429–455.CrossRefGoogle Scholar
Portwood, G., de Bruyn Kops, S. & Caulfield, C. 2019 a Asymptotic dynamics of high dynamic range stratified turbulence. Phys. Rev. Lett. 122 (19), 194504.CrossRefGoogle ScholarPubMed
Portwood, G. D., Mitra, P. P., Ribeiro, M. D., Nguyen, T. M., Nadiga, B. T., Saenz, J. A., Chertkov, M., Garg, A., Anandkumar, A., Dengel, A., et al. 2019 b Turbulence forecasting via neural ODE. arXiv:1911.05180.Google Scholar
Raissi, M., Perdikaris, P. & Karniadakis, G. E. 2017 Physics informed deep learning (Part I): data-driven solutions of nonlinear partial differential equations. arXiv:1711.10561.Google Scholar
Rao, K. J. & de Bruyn Kops, S. M. 2011 A mathematical framework for forcing turbulence applied to horizontally homogeneous stratified flow. Phys. Fluids 23, 065110.CrossRefGoogle Scholar
Sagaut, P. 2006 Large Eddy Simulation for Incompressible Flows, 3rd edn. Springer.Google Scholar
Salehipour, H. & Peltier, W. R. 2019 Deep learning of mixing by two ‘atoms’ of stratified turbulence. J. Fluid Mech. 861, R4.CrossRefGoogle Scholar
Sarghini, F., De Felice, G. & Santini, S. 2003 Neural networks based subgrid scale modeling in large eddy simulations. Comput. Fluids 32 (1), 97–108.CrossRefGoogle Scholar
Shete, K. P. & de Bruyn Kops, S. M. 2019 Area of scalar isosurfaces in homogeneous isotropic turbulence as a function of Reynolds and Schmidt numbers. J. Fluid Mech. 883.Google Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weath. Rev. 91, 99–164.2.3.CO;2>CrossRefGoogle Scholar
Speziale, C. G. 1985 Galilean invariance of subgrid-scale stress models in the large-eddy simulation of turbulence. J. Fluid Mech. 156, 55–62.CrossRefGoogle Scholar
Stolz, S. & Adams, N. A. 1999 An approximate deconvolution procedure for large-eddy simulation. Phys. Fluids 11 (7), 1699–1701.CrossRefGoogle Scholar
Vollant, A., Balarac, G. & Corre, C. 2016 A dynamic regularized gradient model of the subgrid-scale stress tensor for large-eddy simulation. Phys. Fluids 28, 025114.CrossRefGoogle Scholar
Warhaft, Z. 2000 Passive scalar in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.CrossRefGoogle Scholar

Portwood et al. supplementary movie

A visualization of the evolution of passive scalar concentration, and associated local errors, in a-posteriori testing of residual flux models with Δ*=18.
Download Portwood et al. supplementary movie(Video)
Video 26.1 MB