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Asymptotic-preserving neural networks for multiscale Vlasov-Poisson-Fokker-Planck system in the high-field regime. (English) Zbl 07841555

The aim in this article is to develop and investigate the performance of two numerical methods based on Physics-Informed Neural Networks (PINNs), the so-called Asymptotic-Preserving Neural Network (APNN) methods, for solving the multiscale uncertain Vlasov-Poisson-Fokker-Planck system in the high-field regimes. The second section starts with the presentation of the Vlasov-Poisson-Fokker-Planck (VPNN) system from which, taking the limit as \(\epsilon\to 0\), one obtains the high-field limit of the VPFP system: \[ \partial _t \rho - \nabla _x. (\rho\nabla _x \Phi) =0 \] \[ -\Delta_x \Phi _x = \rho - h(x) \] Further the micro-macro model is introduced and one shows the way high-field limit model can be derived from the micro-macro model. In the third section two formulations of the Asymptotic- Preserving Neural Networks (APNNs) methods are described. The one APNN method is based on the micro-macro decomposition for the VPFP system which means to use PINN to solve the micro-macro decomposition system, rather that the original one. The second APNN method is based on the mass conservation low and one shows that the applicability domain for this method is broader and involves cases with long time duration on non-equilibrium data. The performances of both methods are tested, compared and efficiency discussed on a set of very interesting problems and models. In the fourth section there is an extensive presentation of numerical experiments and results for following problems: the Landau Damping, the Bump-on-Tail Case, a one-dimensional Riemann Problem, the case of Mixing Regimes, the Gravitational Case, Uncertainty Quantification (UQ) problems. Some conclusions can be found in the fifth section.

MSC:

82D10 Statistical mechanics of plasmas
82B40 Kinetic theory of gases in equilibrium statistical mechanics
68T07 Artificial neural networks and deep learning
82C32 Neural nets applied to problems in time-dependent statistical mechanics
82M20 Finite difference methods applied to problems in statistical mechanics
35B40 Asymptotic behavior of solutions to PDEs
65D32 Numerical quadrature and cubature formulas
65K10 Numerical optimization and variational techniques
60J65 Brownian motion
35Q83 Vlasov equations

Software:

DGM; DeepONet; PDE-Net

References:

[1] Arnold, A.; Carrillo, JA; Gamba, I.; Shu, C-W, Low and high field scaling limits for the Vlasov-and Wigner-Poisson-Fokker-Planck systems, Transp. Theory Stat. Phys., 30, 121-153, 2001 · Zbl 0987.00055 · doi:10.1081/TT-100105365
[2] Berman, P.; Haverkort, J.; Woerdman, J., Collision kernels and transport coefficients, Phys. Rev. A, 34, 4647, 1986 · doi:10.1103/PhysRevA.34.4647
[3] Bird, GA, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, 1994, Oxford: Clarendon Press, Oxford · doi:10.1093/oso/9780198561958.001.0001
[4] Carrillo, JA; Wang, L.; Xu, W.; Yan, M., Variational asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck system, Multisc. Model. Simul., 19, 478-505, 2021 · Zbl 1467.82071 · doi:10.1137/20M1350431
[5] Cercignani, C.; Gamba, I.; Levermore, C., High field approximations to a Boltzmann-Poisson system and boundary conditions in a semiconductor, Appl. Math. Lett., 10, 111-117, 1997 · Zbl 0894.76072 · doi:10.1016/S0893-9659(97)00069-4
[6] Chandrasekhar, S., Stochastic problems in physics and astronomy, Rev. Mod. Phys., 15, 1, 1943 · Zbl 0061.46403 · doi:10.1103/RevModPhys.15.1
[7] Chen, T.; Chen, H., Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems, IEEE Trans. Neural Netw., 6, 911-917, 1995 · doi:10.1109/72.392253
[8] Chen, Z.; Liu, L.; Mu, L., Solving the linear transport equation by a deep neural network approach, Discrete Contin. Dynam. Syst. - S, 15, 669-686, 2022 · Zbl 1484.65213 · doi:10.3934/dcdss.2021070
[9] Crouseilles, N.; Lemou, M., An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, Kinet. Relat. Models, 4, 441, 2011 · Zbl 1222.82077 · doi:10.3934/krm.2011.4.441
[10] Degond, P.: Asymptotic-Preserving Schemes for Fluid Models of Plasmas, Panoramas et Synthèses, (2013) · Zbl 1308.76182
[11] Delgadillo, R.A., Hu, J., Yang, H.: Multiscale and nonlocal learning for PDEs using densely connected RNNs, (2021) arXiv preprint arXiv:2109.01790,
[12] Goudon, T.; Nieto, J.; Poupaud, F.; Soler, J., Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system, J. Differ. Equ., 213, 418-442, 2005 · Zbl 1072.35176 · doi:10.1016/j.jde.2004.09.008
[13] Havlak, KJ; Victory, HD Jr, The numerical analysis of random particle methods applied to Vlasov-Poisson-Fokker-Planck kinetic equations, SIAM J. Numer. Anal., 33, 291-317, 1996 · Zbl 0864.76100 · doi:10.1137/0733016
[14] Jin, S.: Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Lecture notes for summer school on methods and models of kinetic theory (M &MKT), Porto Ercole, pp. 177-216. Italy, Grosseto (2010) · Zbl 1259.82079
[15] Jin, S., Asymptotic-preserving schemes for multiscale physical problems, Acta Numer., 31, 415-489, 2022 · Zbl 1519.65046 · doi:10.1017/S0962492922000010
[16] Jin, S.; Liu, J-G; Ma, Z., Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro-macro decomposition-based asymptotic-preserving method, Res. Math. Sci., 4, 15, 2017 · Zbl 1375.65004 · doi:10.1186/s40687-017-0105-1
[17] Jin, S., Ma, Z., Wu, K.: Asymptotic-preserving neural networks for multiscale kinetic equations, (2023) arXiv preprint arXiv:2306.15381
[18] Jin, S.; Pareschi, L., Uncertainty Quantification for Hyperbolic and Kinetic Equations, 2017, Berlin: Springer, Berlin · Zbl 1393.35003 · doi:10.1007/978-3-319-67110-9
[19] Jin, S.; Wang, L., An asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck system in the high field regime, Acta Math. Sci., 31, 2219-2232, 2011 · Zbl 1265.82006 · doi:10.1016/S0252-9602(11)60395-0
[20] Jin, S.; Wang, L., Asymptotic-preserving numerical schemes for the semiconductor Boltzmann equation efficient in the high field regime, SIAM J. Sci. Comput., 35, B799-B819, 2013 · Zbl 1277.82047 · doi:10.1137/120886534
[21] Khoo, Y.; Lu, J.; Ying, L., Solving parametric PDE problems with artificial neural networks, Eur. J. Appl. Math., 32, 421-435, 2021 · Zbl 1501.65154 · doi:10.1017/S0956792520000182
[22] Koura, K.; Matsumoto, H., Variable soft sphere molecular model for inverse-power-law or Lennard-Jones potential, Phys. Fluids A, 3, 2459-2465, 1991 · Zbl 0825.76715 · doi:10.1063/1.858184
[23] Kovachki, N.; Li, Z.; Liu, B.; Azizzadenesheli, K.; Bhattacharya, K.; Stuart, A.; Anandkumar, A., Neural operator: learning maps between function spaces with applications to PDEs, J. Mach. Learn. Res., 24, 1-97, 2023
[24] Lagaris, IE; Likas, A.; Fotiadis, DI, Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Netw., 9, 987-1000, 1998 · doi:10.1109/72.712178
[25] Lemou, M.; Mieussens, L., A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31, 334-368, 2008 · Zbl 1187.82110 · doi:10.1137/07069479X
[26] Li, H., Jiang, S., Sun, W., Xu, L., Zhou, G.: A model-data asymptotic-preserving neural network method based on micro-macro decomposition for gray radiative transfer equations, (2022) arXiv preprint arXiv:2212.05523
[27] Li, Z.; Dong, B.; Wang, Y., Learning invariance preserving moment closure model for Boltzmann-BGK equation, Commun. Math. Stat., 11, 59-101, 2023 · Zbl 1512.35430
[28] Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., Anandkumar, A.: Fourier neural operator for parametric partial differential equations, (2020) arXiv preprint arXiv:2010.08895
[29] Li, Z., Zheng, H., Kovachki, N., Jin, D., Chen, H., Liu, B., Azizzadenesheli, K., Anandkumar, A.: Physics-informed neural operator for learning partial differential equations, (2021) arXiv preprint arXiv:2111.03794
[30] Long, Z., Lu, Y., Ma, X., Dong, B.: PDE-Net: Learning PDEs from data. In International Conference on Machine Learning, PMLR, pp. 3208-3216 (2018)
[31] Longo, S., Monte Carlo models of electron and ion transport in non-equilibrium plasmas, Plasma Sour. Sci. Technol., 9, 468, 2000 · doi:10.1088/0963-0252/9/4/303
[32] Lou, Q.; Meng, X.; Karniadakis, GE, Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation, J. Comput. Phys., 447, 2021 · Zbl 07516434 · doi:10.1016/j.jcp.2021.110676
[33] Lu, L.; Jin, P.; Pang, G.; Zhang, Z.; Karniadakis, GE, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nat. Mach. Intell., 3, 218-229, 2021 · doi:10.1038/s42256-021-00302-5
[34] Lu, Y.; Wang, L.; Xu, W., Solving multiscale steady radiative transfer equation using neural networks with uniform stability, Res. Math. Sci., 9, 45, 2022 · Zbl 1492.65286 · doi:10.1007/s40687-022-00345-z
[35] McClenny, L., Braga-Neto, U.:, Self-adaptive physics-informed neural networks using a soft attention mechanism, (2020) arXiv preprint arXiv:2009.04544
[36] Nieto, J.; Poupaud, F.; Soler, J., High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal., 158, 29-59, 2001 · Zbl 1038.82068 · doi:10.1007/s002050100139
[37] Poëtte, G., A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation, J. Comput. Phys., 385, 135-162, 2019 · Zbl 1451.65006 · doi:10.1016/j.jcp.2019.01.052
[38] Poëtte, G., Numerical analysis of the Monte-Carlo noise for the resolution of the deterministic and uncertain linear Boltzmann equation (comparison of non-intrusive gPC and MC-gPC), Journal of Computational and Theoretical, Transport, 51, 1-53, 2022 · Zbl 07585132
[39] Poupaud, F.; Soler, J., Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 10, 1027-1045, 2000 · Zbl 1018.76048 · doi:10.1142/S0218202500000525
[40] Raissi, M.; Perdikaris, P.; Karniadakis, GE, Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378, 686-707, 2019 · Zbl 1415.68175 · doi:10.1016/j.jcp.2018.10.045
[41] Sirignano, J.; Spiliopoulos, K., DGM: a deep learning algorithm for solving partial differential equations, J. Comput. Phys., 375, 1339-1364, 2018 · Zbl 1416.65394 · doi:10.1016/j.jcp.2018.08.029
[42] Stein, M., Large sample properties of simulations using latin hypercube sampling, Technometrics, 29, 143-151, 1987 · Zbl 0627.62010 · doi:10.1080/00401706.1987.10488205
[43] Subramanian, S., Kirby, R.M., Mahoney, M.W., Gholami, A.: Adaptive self-supervision algorithms for physics-informed neural networks, (2022) arXiv preprint arXiv:2207.04084
[44] Victory Jr. H.D., O’Dwyer, B.P.: On classical solutions of Vlasov-Poisson Fokker-Planck systems, Indiana Univ. Math. J. 105-156 (1990) · Zbl 0674.60097
[45] Wang, S.; Wang, H.; Perdikaris, P., Learning the solution operator of parametric partial differential equations with physics-informed DeepONets, Sci. Adv., 7, eabi8605, 2021 · doi:10.1126/sciadv.abi8605
[46] Weinan, E.; Han, J.; Jentzen, A., Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning, Nonlinearity, 35, 278, 2021 · Zbl 1490.60202
[47] Xu, Z.-Q.J., Zhang, Y., Luo, T., Xiao, Y., Ma, Z.: Frequency principle: Fourier analysis sheds light on deep neural networks, (2019) arXiv preprint arXiv:1901.06523
[48] Yu, B.; E, W., The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems, Commun. Math. Stat., 6, 1-12, 2018 · Zbl 1392.35306 · doi:10.1007/s40304-018-0127-z
[49] Zhu, Y.; Jin, S., The Vlasov-Poisson-Fokker-Planck system with uncertainty and a one-dimensional asymptotic preserving method, Multisc. Model. Simul., 15, 1502-1529, 2017 · Zbl 1378.82047 · doi:10.1137/16M1090028
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