Learning Thermodynamically Stable and Galilean Invariant Partial Differential Equations for Non-Equilibrium Flows
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Juntao Huang
Abstract
In this work, we develop a method for learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) based on the conservation-dissipation formalism of irreversible thermodynamics. As governing equations for non-equilibrium flows in one dimension, the learned PDEs are parameterized by fully connected neural networks and satisfy the conservation-dissipation principle automatically. In particular, they are hyperbolic balance laws and Galilean invariant. The training data are generated from a kinetic model with smooth initial data. Numerical results indicate that the learned PDEs can achieve good accuracy in a wide range of Knudsen numbers. Remarkably, the learned dynamics can give satisfactory results with randomly sampled discontinuous initial data and Sod’s shock tube problem although it is trained only with smooth initial data.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12071246
Funding statement: This work was supported by the National Natural Science Foundation of China (Grant No. 12071246).
Acknowledgment
JH would like to thank Qi Tang in Los Alamos National Laboratory for helpful discussions in training of the neural networks.
References
[1] W. -A. Yong, Intrinsic properties of conservation-dissipation formalism of irreversible thermodynamics, Phil. Trans. R. Soc. A 378 (2020), no. 2170, 20190177.10.1098/rsta.2019.0177Suche in Google Scholar PubMed
[2] S. R. De Groot and P. Mazur, Non-equilibrium Thermodynamics, Courier Corporation, 2013.Suche in Google Scholar
[3] I. Müller and T. Ruggeri, Rational Extended Thermodynamics, 1998.10.1007/978-1-4612-2210-1Suche in Google Scholar
[4] D. Jou, J. Casas-Vázquez and G. Lebon, Extended Irreversible Thermodynamics, in: Extended Irreversible Thermodynamics, Springer (1996), 41–74.10.1007/978-3-642-97671-1_2Suche in Google Scholar
[5] G. Lebon, D. Jou and J. Casas-Vázquez, Understanding Non-equilibrium Thermodynamics, volume 295, Springer, 2008.10.1007/978-3-540-74252-4Suche in Google Scholar
[6] H. C. Öttinger, Beyond equilibrium thermodynamics, John Wiley & Sons, 2005.10.1002/0471727903Suche in Google Scholar
[7] M. Pavelka, V. Klika and M. Grmela, Multiscale Thermo-dynamics: Introduction to GENERIC, Walter de Gruyter GmbH & Co KG, 2018.10.1515/9783110350951Suche in Google Scholar
[8] Y. Zhu, L. Hong, Z. Yang and W. -A. Yong, Conservation-dissipation formalism of irreversible thermodynamics, J. Non-Equilib. Thermodyn. 40 (2015), no. 2, 67–74.10.1515/jnet-2014-0037Suche in Google Scholar
[9] Y. Hyon, D. Y. Kwak and C. Liu, Energetic variational approach in complex fluids: maximum dissipation principle, Discrete Contin. Dyn. Syst., Ser. A 26 (2010), no. 4, 1291.10.3934/dcds.2010.26.1291Suche in Google Scholar
[10] W. Muschik, Contact temperature and internal variables: A glance back, 20 years later, J. Non-Equilib. Thermodyn. 39 (2014), no. 3, 113–121.10.1515/jnet-2014-0016Suche in Google Scholar
[11] W. -A. Yong, An interesting class of partial differential equations, J. Math. Phys. 49 (2008), no. 3, 033503.10.1063/1.2884710Suche in Google Scholar
[12] G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27 (1978), no. 1, 1–31.10.1016/0021-9991(78)90023-2Suche in Google Scholar
[13] J. Han, C. Ma, Z. Ma and W. E, Uniformly accurate machine learning-based hydrodynamic models for kinetic equations, Proc. Natl. Acad. Sci. USA 116 (2019), no. 44, 21983–21991.10.1073/pnas.1909854116Suche in Google Scholar PubMed PubMed Central
[14] J. Ling, A. Kurzawski and J. Templeton, Reynolds averaged turbulence modelling using deep neural networks with embedded invariance, J. Fluid Mech. 807 (2016), 155–166.10.1017/jfm.2016.615Suche in Google Scholar
[15] H. Lei, L. Wu and W. E, Machine learning based non-Newtonian fluid model with molecular fidelity, preprint (2020), https://arxiv.org/abs/2003.03672.10.1103/PhysRevE.102.043309Suche in Google Scholar
[16] M. Raissi and G. E. Karniadakis, Hidden physics models: Machine learning of nonlinear partial differential equations, J. Comput. Phys. 357 (2018), 125–141.10.1016/j.jcp.2017.11.039Suche in Google Scholar
[17] M. Raissi, P. Perdikaris and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378 (2019), 686–707.10.1016/j.jcp.2018.10.045Suche in Google Scholar
[18] X. Chen, L. Yang, J. Duan and G. E. Karniadakis, Solving inverse stochastic problems from discrete particle observations using the Fokker-Planck equation and physics-informed neural networks, preprint (2020), https://arxiv.org/abs/2008.10653.Suche in Google Scholar
[19] E. Zhang, M. Yin and G. E. Karniadakis, Physics-informed neural networks for nonhomogeneous material identification in elasticity imaging, preprint (2020), https://arxiv.org/abs/2009.04525.Suche in Google Scholar
[20] M. Yin, X. Zheng, J. D. Humphrey and G. E. Karniadakis, Non-invasive inference of thrombus material properties with physics-informed neural networks, preprint (2020), https://arxiv.org/abs/2005.11380.10.1016/j.cma.2020.113603Suche in Google Scholar
[21] S. L. Brunton, J. L. Proctor and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA 113 (2016), no. 15, 3932–3937.10.1073/pnas.1517384113Suche in Google Scholar PubMed PubMed Central
[22] S. Rudy, A. Alla, S. L. Brunton and J. N. Kutz, Data-driven identification of parametric partial differential equations, SIAM J. Appl. Dyn. Syst. 18 (2019), no. 2, 643–660.10.1137/18M1191944Suche in Google Scholar
[23] Z. Long, Y. Lu, X. Ma and B. Dong, PDE-Net: Learning PDEs from data, in: International Conference on Machine Learning (2018), 3208–3216.Suche in Google Scholar
[24] Z. Long, Y. Lu and B. Dong, PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network, J. Comput. Phys. 399 (2019), 108925.10.1016/j.jcp.2019.108925Suche in Google Scholar
[25] P. Jin, A. Zhu, G. E. Karniadakis and Y. Tang, Symplectic networks: Intrinsic structure-preserving networks for identifying Hamiltonian systems, preprint (2020), https://arxiv.org/abs/2001.03750.10.1016/j.neunet.2020.08.017Suche in Google Scholar
[26] J. W. Burby, Q. Tang and R. Maulik, Fast neural Poincaré maps for toroidal magnetic fields, preprint (2020), https://arxiv.org/abs/2007.04496.10.2172/1637687Suche in Google Scholar
[27] S. Greydanus, M. Dzamba and J. Yosinski, Hamiltonian neural networks, in: Advances in Neural Information Processing Systems (2019), 15379–15389.Suche in Google Scholar
[28] P. Toth, D. J. Rezende, A. Jaegle, S. Racanière, A. Botev and I. Higgins, Hamiltonian generative networks, preprint (2019), https://arxiv.org/abs/1909.13789.Suche in Google Scholar
[29] Y. D. Zhong, B. Dey and A. Chakraborty, Symplectic ODE-net: Learning Hamiltonian dynamics with control, preprint (2019), https://arxiv.org/abs/1909.12077.Suche in Google Scholar
[30] J. Z. Kolter and G. Manek, Learning stable deep dynamics models, in: Advances in Neural Information Processing Systems (2019), 11128–11136.Suche in Google Scholar
[31] H. Yu, X. Tian, W. E and Q. Li, OnsagerNet: Learning stable and interpretable dynamics using a generalized Onsager principle, preprint (2020), https://arxiv.org/abs/2009.02327.10.1103/PhysRevFluids.6.114402Suche in Google Scholar
[32] J. Han, L. Zhang, R. Car and W. E, Deep potential: A general representation of a many-body potential energy surface, preprint (2017), https://arxiv.org/abs/1707.01478.Suche in Google Scholar
[33] L. Zhang, J. Han, H. Wang, R. Car and W. E, Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics, Phys. Rev. Lett. 120 (2018), no. 14, 143001.10.1103/PhysRevLett.120.143001Suche in Google Scholar PubMed
[34] H. Wang, L. Zhang, J. Han and W. E, Deepmd-kit: A deep learning package for many-body potential energy representation and molecular dynamics, Comput. Phys. Commun. 228 (2018), 178–184.10.1016/j.cpc.2018.03.016Suche in Google Scholar
[35] L. Zhang, J. Han, H. Wang, W. Saidi, R. Car and W. E, End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems, in: Advances in Neural Information Processing Systems (2018), 4436–4446.Suche in Google Scholar
[36] J. Han, L. Zhang and W. E, Integrating machine learning with physics-based modeling, preprint (2020), https://arxiv.org/abs/2006.02619.Suche in Google Scholar
[37] K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Natl. Acad. Sci. USA 68 (1971), no. 8, 1686–1688.10.1007/978-1-4612-5385-3_30Suche in Google Scholar
[38] S. K. Godunov, An interesting class of quasilinear systems, in: Dokl. Acad. Nauk SSSR, volume 139 (1961), 521–523.Suche in Google Scholar
[39] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, volume 3, Springer, 2005.10.1007/3-540-29089-3Suche in Google Scholar
[40] H. Struchtrup, Macroscopic transport equations for rarefied gas flows, in: Macroscopic Transport Equations for Rarefied Gas Flows, Springer (2005), 145–160.10.1007/3-540-32386-4_9Suche in Google Scholar
[41] A. G. Baydin, B. A. Pearlmutter, A. A. Radul and J. M. Siskind, Automatic differentiation in machine learning: a survey, J. Mach. Learn. Res. 18 (2017), no. 1, 5595–5637.Suche in Google Scholar
[42] B. Amos, L. Xu and J. Z. Kolter, Input convex neural networks, in: International Conference on Machine Learning (2017), 146–155.Suche in Google Scholar
[43] G. -S. Jiang and C. -W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996), no. 1, 202–228.10.1006/jcph.1996.0130Suche in Google Scholar
[44] U. M. Ascher, S. J. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25 (1997), no. 2-3, 151–167.10.1016/S0168-9274(97)00056-1Suche in Google Scholar
[45] X. -D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1994), no. 1, 200–212.10.1006/jcph.1994.1187Suche in Google Scholar
[46] S. Ruder, An overview of gradient descent optimization algorithms, preprint (2016), https://arxiv.org/abs/1609.04747.Suche in Google Scholar
[47] Z. Cai, Y. Fan, R. Li and Z. Qiao, Dimension-reduced hyperbolic moment method for the Boltzmann equation with BGK-type collision, Commun. Comput. Phys. 15 (2014), no. 5, 1368–1406.10.4208/cicp.220313.281013aSuche in Google Scholar
[48] Z. Cai and R. Li, Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation, SIAM J. Sci. Comput. 32 (2010), no. 5, 2875–2907.10.1137/100785466Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Research Articles
- Non-Equilibrium Molecular Dynamics Study of the Influence of Branching on the Soret Coefficient of Binary Mixtures of Heptane Isomers
- On the Physical Background of Nerve Pulse Propagation: Heat and Energy
- Learning Thermodynamically Stable and Galilean Invariant Partial Differential Equations for Non-Equilibrium Flows
- Continuum Modeling Perspectives of Non-Fourier Heat Conduction in Biological Systems
- The Maximum Power Cycle Operating Between a Heat Source and Heat Sink with Finite Heat Capacities
- Size Effects and Beyond-Fourier Heat Conduction in Room-Temperature Experiments
- The Role of Internal Irreversibilities in the Performance and Stability of Power Plant Models Working at Maximum ϵ-Ecological Function
Artikel in diesem Heft
- Frontmatter
- Research Articles
- Non-Equilibrium Molecular Dynamics Study of the Influence of Branching on the Soret Coefficient of Binary Mixtures of Heptane Isomers
- On the Physical Background of Nerve Pulse Propagation: Heat and Energy
- Learning Thermodynamically Stable and Galilean Invariant Partial Differential Equations for Non-Equilibrium Flows
- Continuum Modeling Perspectives of Non-Fourier Heat Conduction in Biological Systems
- The Maximum Power Cycle Operating Between a Heat Source and Heat Sink with Finite Heat Capacities
- Size Effects and Beyond-Fourier Heat Conduction in Room-Temperature Experiments
- The Role of Internal Irreversibilities in the Performance and Stability of Power Plant Models Working at Maximum ϵ-Ecological Function
