Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-08T16:33:16.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Data-driven discovery of governing equations for fluid dynamics based on molecular simulation

Published online by Cambridge University Press:  31 March 2020

Jun Zhang Open the ORCID record for Jun Zhang [Opens in a new window]  and
Wenjun Ma
Jun Zhang*
Affiliation:
School of Aeronautic Science and Engineering, Beihang University, Beijing100191, PR China
Wenjun Ma
Affiliation:
School of Aeronautic Science and Engineering, Beihang University, Beijing100191, PR China
*
† Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The discovery of governing equations from data is revolutionizing the development of some research fields, where the scientific data are abundant but the well-characterized quantitative descriptions are probably scarce. In this work, we propose to combine the direct simulation Monte Carlo (DSMC) method, which is a popular molecular simulation tool for gas flows, and machine learning to discover the governing equations for fluid dynamics. The DSMC method does not assume any macroscopic governing equations a priori but just relies on the model of molecular interactions at the microscopic level. The data generated by DSMC are utilized to derive the underlying governing equations using a sparse regression method proposed recently. We demonstrate that this strategy is capable of deriving a variety of equations in fluid dynamics, such as the momentum equation, diffusion equation, Fokker–Planck equation and vorticity transport equation. The data-driven discovery not only provides the right forms of the governing equations, but also determines accurate values of the transport coefficients such as viscosity and diffusivity. This work proves that data-driven discovery combined with molecular simulations is a promising and alternative method to derive governing equations in fluid dynamics, and it is expected to pave a new way to establish the governing equations of non-equilibrium flows and complex fluids.

JFM classification

Rarefied Gas Flow: Molecular dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 892 , 10 June 2020 , A5
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alexander, F. J., Garcia, A. L. & Alder, B. J. 1998 Cell size dependence of transport coefficients in stochastic particle algorithms. Phys. Fluids 10 (6), 1540–1542.CrossRefGoogle Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press.Google Scholar
Bongard, J. & Lipson, H. 2007 Automated reverse engineering of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 104 (24), 9943–9948.CrossRefGoogle ScholarPubMed
Brunton, S. L. & Kutz, J. N. 2019 Data-driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press.CrossRefGoogle Scholar
Brunton, S. L., Noack, B. R. & Koumoutsakos, P. 2020 Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508.CrossRefGoogle 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
Chapman, S. & Cowling, T. G. 1990 The Mathematical Theory of Non-uniform Gases. Cambridge University Press.Google Scholar
Daniels, B. C. & Nemenman, I. 2015 Automated adaptive inference of phenomenological dynamical models. Nat. Commun. 6, 8133.CrossRefGoogle ScholarPubMed
Duraisamy, K., Iaccarino, G. & Xiao, H. 2019 Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51, 357–377.CrossRefGoogle Scholar
Gallis, M. A., Bitter, N. P., Koehler, T. P., Torczynski, J. R., Plimpton, S. J. & Papadakis, G. 2017 Molecular-level simulations of turbulence and its decay. Phys. Rev. Lett. 118 (6), 064501.CrossRefGoogle ScholarPubMed
Garcia, A. L. & Wagner, W. 2000 Time step truncation error in direct simulation Monte Carlo. Phys. Fluids 12 (10), 2621–2633.CrossRefGoogle Scholar
Gonzalez, G. R., Rico, M. R. & Kevrekidis, I. G. 1998 Identification of distributed parameter systems: a neural net based approach. Comput. Chem. Engng 22, S965–S968.CrossRefGoogle Scholar
Gu, X. J., Barber, R. W., John, B. & Emerson, D. R. 2019 Non-equilibrium effects on flow past a circular cylinder in the slip and early transition regime. J. Fluid Mech. 860, 654–681.CrossRefGoogle Scholar
Gu, X. J. & Emerson, D. R. 2009 A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. J. Fluid Mech. 636, 177–216.CrossRefGoogle Scholar
Hadjiconstantinou, N. G., Garcia, A. L., Bazant, M. Z. & He, G. 2003 Statistical error in particle simulations of hydrodynamic phenomena. J. Comput. Phys. 187 (1), 274–297.CrossRefGoogle Scholar
Hey, J. G., Tansley, S. & Tolle, K. M. 2009 The Fourth Paradigm: Data-intensive Scientific Discovery, vol. 1. Microsoft Research.Google Scholar
Jordan, M. I. & Mitchell, T. M. 2015 Machine learning: trends, perspectives, and prospects. Science 349 (6245), 255–260.CrossRefGoogle ScholarPubMed
Kevrekidis, I. G., Gear, C. W., Hyman, J. M., Kevrekidid, P. G., Runborg, O. & Theodoropoulos, C. 2003 Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Commun. Math. Sci. 1 (4), 715–762.Google Scholar
Lockerby, D. A., Reese, J. M. & Gallis, M. A. 2005 Capturing the Knudsen layer in continuum-fluid models of nonequilibrium gas flows. AIAA J. 43 (6), 1391–1393.CrossRefGoogle Scholar
Loiseau, J.-C. & Brunton, S. L. 2018 Constrained sparse Galerkin regression. J. Fluid Mech. 838, 42–67.CrossRefGoogle Scholar
Manela, A. & Zhang, J. 2012 The effect of compressibility on the stability of wall-bounded Kolmogorov flow. J. Fluid Mech. 694, 29–49.CrossRefGoogle Scholar
Marx, V. 2013 The big challenges of big data. Nature 498, 255–260.CrossRefGoogle ScholarPubMed
Oran, E. S., Oh, C. K. & Cybyk, B. Z. 1998 Direct simulation Monte Carlo: recent advances and applications. Annu. Rev. Fluid Mech. 30 (1), 403–441.CrossRefGoogle Scholar
Rudy, S., Alla, A., Brunton, S. L. & Kutz, J. N. 2019 Data-driven identification of parametric partial differential equations. SIAM J. Appl. Dyn. Syst. 18 (2), 643–660.CrossRefGoogle Scholar
Rudy, S. H., Brunton, S. L., Proctor, J. L. & Kutz, J. N. 2017 Data-driven discovery of partial differential equations. Sci. Adv. 3 (4), e1602614.CrossRefGoogle ScholarPubMed
Schaeffer, H. 2017 Learning partial differential equations via data discovery and sparse optimization. Proc. R. Soc. Lond. A 473 (2197), 20160446.CrossRefGoogle ScholarPubMed
Schmidt, M. & Lipson, H. 2009 Distilling free-form natural laws from experimental data. Science 324 (5923), 81–85.CrossRefGoogle ScholarPubMed
Stefanov, S., Roussinov, V. & Cercignani, C. 2002 Rayleigh–Bénard flow of a rarefied gas and its attractors. I. Convection regime. Phys. Fluids 14 (7), 2255–2269.CrossRefGoogle Scholar
Struchtrup, H. 2005 Macroscopic Transport Equations for Rarefied Gas Flows. Springer.CrossRefGoogle Scholar
Struchtrup, H. & Torrilhon, M. 2003 Regularization of Grads 13 moment equations: derivation and linear analysis. Phys. Fluids 15 (9), 2668–2680.CrossRefGoogle Scholar
Sun, Q. H. & Boyd, I. D. 2002 A direct simulation method for subsonic, microscale gas flows. J. Comput. Phys. 179 (2), 400–425.CrossRefGoogle Scholar
Tibshirani, R. 1996 Regression shrinkage and selection via the lasso. J. R. Stat. Soc. B 58 (1), 267–288.Google Scholar
Wagner, W. 1992 A convergence proof for Bird’s direct simulation Monte Carlo method for the Boltzmann equation. J. Stat. Phys. 66 (3–4), 1011–1044.CrossRefGoogle Scholar
Zhang, J. & Fan, J. 2009 Monte Carlo simulation of thermal fluctuations below the onset of Rayleigh–Bénard convection. Phys. Rev. E 79 (5), 056302.Google ScholarPubMed
Zhang, J., Fan, J. & Fei, F. 2010 Effects of convection and solid wall on the diffusion in microscale convection flows. Phys. Fluids 22 (12), 122005.CrossRefGoogle Scholar
Zhang, J., John, B., Pfeiffer, M., Fei, F. & Wen, D. S. 2019a Particle-based hybrid and multiscale methods for nonequilibrium gas flows. Adv. Aerodyn. 1 (1), 12.CrossRefGoogle Scholar
Zhang, J. & Önskog, T. 2017 Langevin equation elucidates the mechanism of the Rayleigh–Bénard instability by coupling molecular motions and macroscopic fluctuations. Phys. Rev. E 96 (4), 043104.Google ScholarPubMed
Zhang, J., Tian, P., Yao, S. Q. & Fei, F. 2019b Multiscale investigation of Kolmogorov flow: from microscopic molecular motions to macroscopic coherent structures. Phys. Fluids 31 (8), 082008.Google Scholar