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A kinetic chemotaxis model with internal states and temporal sensing
A neural network closure for the Euler-Poisson system based on kinetic simulations
1. | Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 Strasbourg, France |
2. | INRIA Nancy-Grand Est, TONUS Project, Strasbourg, France |
This work deals with the modeling of plasmas, which are ionized gases. Thanks to machine learning, we construct a closure for the one-dimensional Euler-Poisson system valid for a wide range of collisional regimes. This closure, based on a fully convolutional neural network called V-net, takes as input the whole spatial density, mean velocity and temperature and predicts as output the whole heat flux. It is learned from data coming from kinetic simulations of the Vlasov-Poisson equations. Data generation and preprocessings are designed to ensure an almost uniform accuracy over the chosen range of Knudsen numbers (which parametrize collisional regimes). Finally, several numerical tests are carried out to assess validity and flexibility of the whole pipeline.
References:
[1] |
A. Beck, D. Flad and C.-D. Munz, Deep neural networks for data-driven les closure models, J. Comput. Phys., 398 (2019), 108910, 23 pp.
doi: 10.1016/j.jcp.2019.108910. |
[2] |
N. Besse, F. Berthelin, Y. Brenier and P. Bertrand,
The multi-water-bag equations for collisionless kinetic modeling, Kinet. Relat. Models, 2 (2009), 39-80.
doi: 10.3934/krm.2009.2.39. |
[3] |
N. Besse and P. Bertrand,
Gyro-water-bag approach in nonlinear gyrokinetic turbulence, J. Comput. Phys., 228 (2009), 3973-3995.
doi: 10.1016/j.jcp.2009.02.025. |
[4] |
S. I. Braginskii,
Transport phenomena in plasma, Rev. Plasma Phys., 1 (1963), 205.
|
[5] |
A. Brizard,
Nonlinear gyrofluid description of turbulent magnetized plasmas, Phys. Fluids B, 4 (1992), 1213-1228.
doi: 10.1063/1.860129. |
[6] |
Z. Cai, Y. Fan and R. Li,
Globally hyperbolic regularization of grad's moment system, Communications on Pure and Applied Mathematics, 67 (2014), 464-518.
doi: 10.1002/cpa.21472. |
[7] |
Z. Chang and J. D. Callen,
Unified fluid/kinetic description of plasma microinstabilities. part i: Basic equations in a sheared slab geometry, Physics of Fluids B: Plasma Physics, 4 (1992), 1167-1181.
doi: 10.1063/1.860125. |
[8] |
G. F. Chew, M. L. Goldberger, F. E. Low and Y. Nambu,
Application of dispersion relations to low-energy meson-nucleon scattering, Phys. Rev., 106 (1957), 1337-1344.
doi: 10.1103/PhysRev.106.1337. |
[9] |
A. Crestetto, N. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles, Kinet. Relat. Models, 5 (2012) 787–816.
doi: 10.3934/krm.2012.5.787. |
[10] |
N. Crouseilles, P. Degond and M. Lemou,
A hybrid kinetic/fluid model for solving the gas dynamics boltzmann–bgk equation, J. Comput. Phys., 199 (2004), 776-808.
doi: 10.1016/j.jcp.2004.03.007. |
[11] |
P. Degond, Macroscopic limits of the boltzmann equation: A review, In Modeling and Computational Methods for Kinetic Equations, Birkhäuser, Boston, MA, 2004, 3–57. |
[12] |
P. Degond, G. Dimarco and L. Mieussens,
A multiscale kinetic–fluid solver with dynamic localization of kinetic effects, J. Comput. Phys., 229 (2010), 4907-4933.
doi: 10.1016/j.jcp.2010.03.009. |
[13] |
O. Desjardins, R. O. Fox and P. Villedieu,
A quadrature-based moment method for dilute fluid-particle flows, J. Comput. Phys., 227 (2008), 2514-2539.
doi: 10.1016/j.jcp.2007.10.026. |
[14] |
G. Dimarco and L. Pareschi,
Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.
doi: 10.1017/S0962492914000063. |
[15] |
B. Dubroca and J.-L. Feugeas, Etude théorique et numérique d'une hiérarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 915–920.
doi: 10.1016/S0764-4442(00)87499-6. |
[16] |
B. Dubroca, J.-L. Feugeas and M. Frank,
Angular moment model for the Fokker-Planck equation, The European Physical Journal D, 60 (2010), 301-207.
doi: 10.1140/epjd/e2010-00190-8. |
[17] |
D. Dumoulin and F. Visin, A guide to convolution arithmetic for deep learning, 2016. |
[18] |
K. Duraisamy, G. Iaccarino and H. Xiao,
Turbulence modeling in the age of data, Annu. Rev. Fluid Mech., 51 (2019), 357-377.
|
[19] |
C. K. Garrett and C. D. Hauck,
A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport Theor. Stat. Phys., 42 (2013), 203-235.
doi: 10.1080/00411450.2014.910226. |
[20] |
H. Grad,
On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[21] |
G. W. Hammett, W. Dorland and F. W. Perkins,
Fluid models of phase mixing, landau damping, and nonlinear gyrokinetic dynamics, Phys. Fluids B, 4 (1992), 2052-2061.
doi: 10.1063/1.860014. |
[22] |
G. W. Hammett and F. W. Perkins,
Fluid moment models for landau damping with application to the ion-temperature-gradient instability, Phys. Rev. Lett., 64 (1990), 3019-3022.
doi: 10.1103/PhysRevLett.64.3019. |
[23] |
J. Han, C. Ma, Z. Ma and W. E,
Uniformly accurate machine learning-based hydrodynamic models for kinetic equations, PNAS, 116 (2019), 21983-21991.
doi: 10.1073/pnas.1909854116. |
[24] |
K. He, X. Zhang, S. Ren and J. Sun, Deep residual learning for image recognition, In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016,770–778.
doi: 10.1109/CVPR.2016.90. |
[25] |
P. Helluy, L. Navoret, N. Pham and A. Crestetto, Reduced Vlasov-Maxwell simulations, C. R. Mécanique, 342 (2014), 619–635.
doi: 10.1016/j.crme.2014.06.008. |
[26] |
P. Hunana, G. P. Zank, M. Laurenza, A. Tenerani, G. M. Webb, M. L. Goldstein, M. Velli and L. Adhikari,
New closures for more precise modeling of landau damping in the fluid framework., Physical Review Letters, 121 (2018), 135101.
doi: 10.1103/PhysRevLett.121.135101. |
[27] |
M. Junk,
Domain of definition of levermore's five-moment system, J. Stat. Phys., 93 (1998), 1143-1167.
doi: 10.1023/B:JOSS.0000033155.07331.d9. |
[28] |
R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, volume 31. Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253. |
[29] |
C. D. Levermore,
Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[30] |
C. D. Levermore and W. J. Morokoff,
The gaussian moment closure for gas dynamics, SIAM J. Appl. Math., 59 (1998), 72-96.
doi: 10.1137/S0036139996299236. |
[31] |
J. L. Lumley,
Toward a turbulent constitutive relation, J. Fluid Mech, 41 (1970), 413-434.
doi: 10.1017/S0022112070000678. |
[32] |
C. Ma, B. Zhu, X.-Q. Xu and W. Wang, Machine learning surrogate models for landau fluid closure, Physics of Plasmas, 27 (2020), 042502, arXiv: 1909.11509.
doi: 10.1063/1.5129158. |
[33] |
G. Manfredi, Density functional theory for collisionless plasmas–equivalence of fluid and kinetic approaches, J. Plasma Phys., 86 (2020).
doi: 10.1017/S0022377820000240. |
[34] |
R. Maulik, N. A. Garland, J. W. Burby, X.-Z. Tang and P. Balaprakash,
Neural network representability of fully ionized plasma fluid model closures, Phys. Plasmas, 27 (2020), 072106.
doi: 10.1063/5.0006457. |
[35] |
F. Milletari, N. Navab and S. Ahmadi, V-net: Fully convolutional neural networks for volumetric medical image segmentation, In 2016 Fourth International Conference on 3D Vision (3DV), 2016,565–571.
doi: 10.1109/3DV.2016.79. |
[36] |
C. Negulescu and S. Possanner,
Closure of the strongly magnetized electron fluid equations in the adiabatic regime, Multiscale Model. Sim., 14 (2016), 839-873.
doi: 10.1137/15M1027309. |
[37] |
T. Passot, P. L. Sulem and P. Hunana,
Extending magnetohydrodynamics to the slow dynamics of collisionless plasmas, Physics of Plasmas, 19 (2012), 082113.
doi: 10.1063/1.4746092. |
[38] |
M. Perin, C. Chandre, P. J. Morrison and E. Tassi,
Hamiltonian closures for fluid models with four moments by dimensional analysis, J. Phys. A Math. Theor., 48 (2015), 275501.
doi: 10.1088/1751-8113/48/27/275501. |
[39] |
N. Pham, P. Helluy and A. Crestetto,
Space-only hyperbolic approximation of the vlasov equation, ESAIM: Proc., 43 (2013), 17-36.
doi: 10.1051/proc/201343002. |
[40] |
O. Ronneberger, P. Fischer and T. Brox, U-net: Convolutional networks for biomedical image segmentation, In International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, 2015,234–241.
doi: 10.1007/978-3-319-24574-4_28. |
[41] |
J. Schneider,
Entropic approximation in kinetic theory, Esaim Math Model Numer Anal., 38 (2004), 541-561.
doi: 10.1051/m2an:2004025. |
[42] |
C. Shorten and T. M. Khoshgoftaar,
A survey on image data augmentation for deep learning, Journal of Big Data, 6 (2019), 60.
doi: 10.1186/s40537-019-0197-0. |
[43] |
P. B. Snyder, G. W. Hammett and W. Dorland,
Landau fluid models of collisionless magnetohydrodynamics, Phys. Plasmas, 4 (1997), 3974-3985.
doi: 10.1063/1.872517. |
[44] |
E. Sonnendrücker, Numerical Methods for the Vlasov-Maxwell Equations, 2015. |
[45] |
H. Struchtrup and M. Torrilhon,
Regularization of grad's 13 moment equations: Derivation and linear analysis, Physics of Fluids, 15 (2003), 2668-2680.
doi: 10.1063/1.1597472. |
[46] |
E. Tassi,
Hamiltonian closures in fluid models for plasmas, Eur. Phys. J. D, 71 (2017), 269.
doi: 10.1140/epjd/e2017-80223-6. |
[47] |
M. Torrilhon,
Hyperbolic moment equations in kinetic gas theory based on multi-variate pearson-iv-distributions, Commun. in Comput. Phys., 7 (2010), 639-673.
doi: 10.4208/cicp.2009.09.049. |
[48] |
M. Torrilhon,
Modeling nonequilibrium gas flow based on moment equations, Annual Review of Fluid Mechanics, 48 (2016), 429-458.
|
[49] |
J.-X. Wang, J.-L. Wu and H. Xiao,
Physics-informed machine learning approach for reconstructing reynolds stress modeling discrepancies based on dns data, Physic. Rev. Fluids, 2 (2017), 034603.
doi: 10.1103/PhysRevFluids.2.034603. |
[50] |
X.-H. Zhou, J. Han and H. Xiao, Learning nonlocal constitutive models with neural networks, Comput. Methods Appl. Mech. Engrg., 384 (2021), Paper No. 113927, 27 pp. arXiv: 2010.10491.
doi: 10.1016/j.cma.2021.113927. |
show all references
References:
[1] |
A. Beck, D. Flad and C.-D. Munz, Deep neural networks for data-driven les closure models, J. Comput. Phys., 398 (2019), 108910, 23 pp.
doi: 10.1016/j.jcp.2019.108910. |
[2] |
N. Besse, F. Berthelin, Y. Brenier and P. Bertrand,
The multi-water-bag equations for collisionless kinetic modeling, Kinet. Relat. Models, 2 (2009), 39-80.
doi: 10.3934/krm.2009.2.39. |
[3] |
N. Besse and P. Bertrand,
Gyro-water-bag approach in nonlinear gyrokinetic turbulence, J. Comput. Phys., 228 (2009), 3973-3995.
doi: 10.1016/j.jcp.2009.02.025. |
[4] |
S. I. Braginskii,
Transport phenomena in plasma, Rev. Plasma Phys., 1 (1963), 205.
|
[5] |
A. Brizard,
Nonlinear gyrofluid description of turbulent magnetized plasmas, Phys. Fluids B, 4 (1992), 1213-1228.
doi: 10.1063/1.860129. |
[6] |
Z. Cai, Y. Fan and R. Li,
Globally hyperbolic regularization of grad's moment system, Communications on Pure and Applied Mathematics, 67 (2014), 464-518.
doi: 10.1002/cpa.21472. |
[7] |
Z. Chang and J. D. Callen,
Unified fluid/kinetic description of plasma microinstabilities. part i: Basic equations in a sheared slab geometry, Physics of Fluids B: Plasma Physics, 4 (1992), 1167-1181.
doi: 10.1063/1.860125. |
[8] |
G. F. Chew, M. L. Goldberger, F. E. Low and Y. Nambu,
Application of dispersion relations to low-energy meson-nucleon scattering, Phys. Rev., 106 (1957), 1337-1344.
doi: 10.1103/PhysRev.106.1337. |
[9] |
A. Crestetto, N. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles, Kinet. Relat. Models, 5 (2012) 787–816.
doi: 10.3934/krm.2012.5.787. |
[10] |
N. Crouseilles, P. Degond and M. Lemou,
A hybrid kinetic/fluid model for solving the gas dynamics boltzmann–bgk equation, J. Comput. Phys., 199 (2004), 776-808.
doi: 10.1016/j.jcp.2004.03.007. |
[11] |
P. Degond, Macroscopic limits of the boltzmann equation: A review, In Modeling and Computational Methods for Kinetic Equations, Birkhäuser, Boston, MA, 2004, 3–57. |
[12] |
P. Degond, G. Dimarco and L. Mieussens,
A multiscale kinetic–fluid solver with dynamic localization of kinetic effects, J. Comput. Phys., 229 (2010), 4907-4933.
doi: 10.1016/j.jcp.2010.03.009. |
[13] |
O. Desjardins, R. O. Fox and P. Villedieu,
A quadrature-based moment method for dilute fluid-particle flows, J. Comput. Phys., 227 (2008), 2514-2539.
doi: 10.1016/j.jcp.2007.10.026. |
[14] |
G. Dimarco and L. Pareschi,
Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.
doi: 10.1017/S0962492914000063. |
[15] |
B. Dubroca and J.-L. Feugeas, Etude théorique et numérique d'une hiérarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 915–920.
doi: 10.1016/S0764-4442(00)87499-6. |
[16] |
B. Dubroca, J.-L. Feugeas and M. Frank,
Angular moment model for the Fokker-Planck equation, The European Physical Journal D, 60 (2010), 301-207.
doi: 10.1140/epjd/e2010-00190-8. |
[17] |
D. Dumoulin and F. Visin, A guide to convolution arithmetic for deep learning, 2016. |
[18] |
K. Duraisamy, G. Iaccarino and H. Xiao,
Turbulence modeling in the age of data, Annu. Rev. Fluid Mech., 51 (2019), 357-377.
|
[19] |
C. K. Garrett and C. D. Hauck,
A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport Theor. Stat. Phys., 42 (2013), 203-235.
doi: 10.1080/00411450.2014.910226. |
[20] |
H. Grad,
On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[21] |
G. W. Hammett, W. Dorland and F. W. Perkins,
Fluid models of phase mixing, landau damping, and nonlinear gyrokinetic dynamics, Phys. Fluids B, 4 (1992), 2052-2061.
doi: 10.1063/1.860014. |
[22] |
G. W. Hammett and F. W. Perkins,
Fluid moment models for landau damping with application to the ion-temperature-gradient instability, Phys. Rev. Lett., 64 (1990), 3019-3022.
doi: 10.1103/PhysRevLett.64.3019. |
[23] |
J. Han, C. Ma, Z. Ma and W. E,
Uniformly accurate machine learning-based hydrodynamic models for kinetic equations, PNAS, 116 (2019), 21983-21991.
doi: 10.1073/pnas.1909854116. |
[24] |
K. He, X. Zhang, S. Ren and J. Sun, Deep residual learning for image recognition, In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016,770–778.
doi: 10.1109/CVPR.2016.90. |
[25] |
P. Helluy, L. Navoret, N. Pham and A. Crestetto, Reduced Vlasov-Maxwell simulations, C. R. Mécanique, 342 (2014), 619–635.
doi: 10.1016/j.crme.2014.06.008. |
[26] |
P. Hunana, G. P. Zank, M. Laurenza, A. Tenerani, G. M. Webb, M. L. Goldstein, M. Velli and L. Adhikari,
New closures for more precise modeling of landau damping in the fluid framework., Physical Review Letters, 121 (2018), 135101.
doi: 10.1103/PhysRevLett.121.135101. |
[27] |
M. Junk,
Domain of definition of levermore's five-moment system, J. Stat. Phys., 93 (1998), 1143-1167.
doi: 10.1023/B:JOSS.0000033155.07331.d9. |
[28] |
R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, volume 31. Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253. |
[29] |
C. D. Levermore,
Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[30] |
C. D. Levermore and W. J. Morokoff,
The gaussian moment closure for gas dynamics, SIAM J. Appl. Math., 59 (1998), 72-96.
doi: 10.1137/S0036139996299236. |
[31] |
J. L. Lumley,
Toward a turbulent constitutive relation, J. Fluid Mech, 41 (1970), 413-434.
doi: 10.1017/S0022112070000678. |
[32] |
C. Ma, B. Zhu, X.-Q. Xu and W. Wang, Machine learning surrogate models for landau fluid closure, Physics of Plasmas, 27 (2020), 042502, arXiv: 1909.11509.
doi: 10.1063/1.5129158. |
[33] |
G. Manfredi, Density functional theory for collisionless plasmas–equivalence of fluid and kinetic approaches, J. Plasma Phys., 86 (2020).
doi: 10.1017/S0022377820000240. |
[34] |
R. Maulik, N. A. Garland, J. W. Burby, X.-Z. Tang and P. Balaprakash,
Neural network representability of fully ionized plasma fluid model closures, Phys. Plasmas, 27 (2020), 072106.
doi: 10.1063/5.0006457. |
[35] |
F. Milletari, N. Navab and S. Ahmadi, V-net: Fully convolutional neural networks for volumetric medical image segmentation, In 2016 Fourth International Conference on 3D Vision (3DV), 2016,565–571.
doi: 10.1109/3DV.2016.79. |
[36] |
C. Negulescu and S. Possanner,
Closure of the strongly magnetized electron fluid equations in the adiabatic regime, Multiscale Model. Sim., 14 (2016), 839-873.
doi: 10.1137/15M1027309. |
[37] |
T. Passot, P. L. Sulem and P. Hunana,
Extending magnetohydrodynamics to the slow dynamics of collisionless plasmas, Physics of Plasmas, 19 (2012), 082113.
doi: 10.1063/1.4746092. |
[38] |
M. Perin, C. Chandre, P. J. Morrison and E. Tassi,
Hamiltonian closures for fluid models with four moments by dimensional analysis, J. Phys. A Math. Theor., 48 (2015), 275501.
doi: 10.1088/1751-8113/48/27/275501. |
[39] |
N. Pham, P. Helluy and A. Crestetto,
Space-only hyperbolic approximation of the vlasov equation, ESAIM: Proc., 43 (2013), 17-36.
doi: 10.1051/proc/201343002. |
[40] |
O. Ronneberger, P. Fischer and T. Brox, U-net: Convolutional networks for biomedical image segmentation, In International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, 2015,234–241.
doi: 10.1007/978-3-319-24574-4_28. |
[41] |
J. Schneider,
Entropic approximation in kinetic theory, Esaim Math Model Numer Anal., 38 (2004), 541-561.
doi: 10.1051/m2an:2004025. |
[42] |
C. Shorten and T. M. Khoshgoftaar,
A survey on image data augmentation for deep learning, Journal of Big Data, 6 (2019), 60.
doi: 10.1186/s40537-019-0197-0. |
[43] |
P. B. Snyder, G. W. Hammett and W. Dorland,
Landau fluid models of collisionless magnetohydrodynamics, Phys. Plasmas, 4 (1997), 3974-3985.
doi: 10.1063/1.872517. |
[44] |
E. Sonnendrücker, Numerical Methods for the Vlasov-Maxwell Equations, 2015. |
[45] |
H. Struchtrup and M. Torrilhon,
Regularization of grad's 13 moment equations: Derivation and linear analysis, Physics of Fluids, 15 (2003), 2668-2680.
doi: 10.1063/1.1597472. |
[46] |
E. Tassi,
Hamiltonian closures in fluid models for plasmas, Eur. Phys. J. D, 71 (2017), 269.
doi: 10.1140/epjd/e2017-80223-6. |
[47] |
M. Torrilhon,
Hyperbolic moment equations in kinetic gas theory based on multi-variate pearson-iv-distributions, Commun. in Comput. Phys., 7 (2010), 639-673.
doi: 10.4208/cicp.2009.09.049. |
[48] |
M. Torrilhon,
Modeling nonequilibrium gas flow based on moment equations, Annual Review of Fluid Mechanics, 48 (2016), 429-458.
|
[49] |
J.-X. Wang, J.-L. Wu and H. Xiao,
Physics-informed machine learning approach for reconstructing reynolds stress modeling discrepancies based on dns data, Physic. Rev. Fluids, 2 (2017), 034603.
doi: 10.1103/PhysRevFluids.2.034603. |
[50] |
X.-H. Zhou, J. Han and H. Xiao, Learning nonlocal constitutive models with neural networks, Comput. Methods Appl. Mech. Engrg., 384 (2021), Paper No. 113927, 27 pp. arXiv: 2010.10491.
doi: 10.1016/j.cma.2021.113927. |


























Hyper-parameter | Value |
size of the input window ( |
512 |
number of levels ( |
5 |
depth ( |
4 |
size of the kernels ( |
11 |
activation function | softplus |
Hyper-parameter | Value |
size of the input window ( |
512 |
number of levels ( |
5 |
depth ( |
4 |
size of the kernels ( |
11 |
activation function | softplus |
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