[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ skip to main content
research-article

Non-diffusive neural network method for hyperbolic conservation laws

Published: 08 August 2024 Publication History

Abstract

In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately computing weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in subdomains bounded by discontinuity lines (DLs), the latter defined from the Rankine-Hugoniot jump conditions. The proposed approach allows to efficiently consider an arbitrary number of entropic shock waves, shock wave generation, as well as wave interactions. Some numerical experiments are presented to illustrate the strengths and properties of the algorithms.

Highlights

Conservation laws.
Neural network solver.
Accurate discontinuity tracking.
Shock wave generation.
Wave interaction.

References

[1]
J.-M. Ghidaglia, F. Pascal, The normal flux method at the boundary for multidimensional finite volume approximations in CFD, Eur. J. Mech. B, Fluids 24 (1) (2005) 1–17.
[2]
E. Godlewski, P.-A. Raviart, Hyperbolic Systems of Conservation Laws, Mathématiques & Applications (Paris), vol. 3/4, Ellipses, Paris, 1991.
[3]
D. Serre, Systèmes de lois de conservation. I, Fondations. [Foundations] Diderot Editeur, Paris, 1996, Hyperbolicité, entropies, ondes de choc. [Hyperbolicity, entropies, shock waves].
[4]
P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002.
[5]
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York-Berlin, 1983.
[6]
E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences., vol. 118, Springer-Verlag, New York, 1996.
[7]
B. Després, Lax theorem and finite volume schemes, Math. Comput. 74 (247) (2004).
[8]
M. Laforest, P.G. LeFloch, Diminishing functionals for nonclassical entropy solutions selected by kinetic relations, Port. Math. 67 (3) (2010).
[9]
I.E. Lagaris, A. Likas, D.I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Netw. 9 (5) (1998) 987–1000.
[10]
M. Raissi, P. Perdikaris, 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.
[11]
G. Pang, L. Lu, G.E. Karniadakis, fPINNs: fractional physics-informed neural networks, SIAM J. Sci. Comput. 41 (4) (2019) A2603–A2626.
[12]
L. Yang, D. Zhang, G.E. Karniadakis, Physics-informed generative adversarial networks for stochastic differential equations, SIAM J. Sci. Comput. 42 (1) (2020) A292–A317.
[13]
J. Han, A. Jentzen, E. Weinan, Solving high-dimensional partial differential equations using deep learning, Proc. Natl. Acad. Sci. USA 115 (34) (2018) 8505–8510.
[14]
H. Gao, M.J. Zahr, J.-X. Wang, Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems, Comput. Methods Appl. Mech. Eng. 390 (2022).
[15]
J. Sirignano, K. Spiliopoulos, DGM: a deep learning algorithm for solving partial differential equations, J. Comput. Phys. 375 (2018) 1339–1364.
[16]
A.D. Jagtap, E. Kharazmi, G.E. Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems, Comput. Methods Appl. Mech. Eng. 365 (2020).
[17]
R. Rodriguez-Torrado, P. Ruiz, L. Cueto-Felgueroso, Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem, Sci. Rep. 12 (2022) 7557.
[18]
R.G. Patel, I. Manickam, N.A. Trask, M.A. Wood, M. Lee, I. Tomas, E.C. Cyr, Thermodynamically consistent physics-informed neural networks for hyperbolic systems, J. Comput. Phys. 449 (2022).
[19]
E. Lorin, X. Yang, Schwarz waveform relaxation-learning for advection-diffusion-reaction equations, J. Comput. Phys. 473 (2023).
[20]
E. Lorin, X. Yang, Neural network-based quasi-optimal domain decomposition method for computing the Schrödinger equation, Comput. Phys. Commun. 299 (2024).
[21]
M. Gander, L. Halpern, Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems, SIAM J. Numer. Anal. 45 (2) (2007) 666–697.
[22]
M.J. Gander, F. Kwok, B.C. Mandal, Dirichlet–Neumann waveform relaxation methods for parabolic and hyperbolic problems in multiple subdomains, BIT Numer. Math. 61 (1) (2021) 173–207.
[23]
M.J. Gander, C. Rohde, Overlapping Schwarz waveform relaxation for convection-dominated nonlinear conservation laws, SIAM J. Sci. Comput. 27 (2) (2006) 415–439.
[24]
X. Antoine, E. Lorin, An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations, Numer. Math. 137 (4) (2017) 923–958.
[25]
L. Bottou, F.E. Curtis, J. Nocedal, Optimization methods for large-scale machine learning, SIAM Rev. 60 (2) (2018) 223–311.
[26]
J. Bradbury, R. Frostig, P. Hawkins, M.J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, Q. Zhang, JAX: Composable Transformations of Python+NumPy Programs, 2018.
[27]
J.L. Montagne, H.C. Yee, M. Vinokur, Comparative study of high-resolution shock-capturing schemes for a real gas, in: M. Deville (Ed.), Proc. Seventh Gamm Conf. on Numerical Methods in Fluid Mechanics, vol. 20, ISBN 3-528-08094-9, 1987.
[28]
L. Halpern, J. Szeftel, Optimized and quasi-optimal Schwarz waveform relaxation for the one-dimensional Schrödinger equation, Math. Models Methods Appl. Sci. 20 (12) (2010) 2167–2199.
[29]
X. Antoine, F. Hou, E. Lorin, Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves, ESAIM: M2AN 52 (4) (2018) 1569–1596.
[30]
X. Antoine, E. Lorin, On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation, J. Comput. Appl. Math. 354 (2019) 15–30.

Recommendations

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of Computational Physics
Journal of Computational Physics  Volume 513, Issue C
Sep 2024
1004 pages

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 08 August 2024

Author Tags

  1. Hyperbolic equations
  2. Conservation laws
  3. Weak solutions
  4. Optimization
  5. Neural network
  6. Scientific machine learning

Qualifiers

  • Research-article

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • 0
    Total Citations
  • 0
    Total Downloads
  • Downloads (Last 12 months)0
  • Downloads (Last 6 weeks)0
Reflects downloads up to 13 Jan 2025

Other Metrics

Citations

View Options

View options

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media