doi: 10.3934/dcdss.2021070
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Solving the linear transport equation by a deep neural network approach

1. 

Department of Mathematics, University of Massachusetts, Dartmouth, MA, 02747

2. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

3. 

Department of Mathematics, University of Georgia, Athens, GA 30602

* Corresponding author: Lin Mu

Received  January 2021 Revised  April 2021 Early access June 2021

Fund Project: Liu is supported by the start-up fund by The Chinese University of Hong Kong

In this paper, we study linear transport model by adopting deep learning method, in particular deep neural network (DNN) approach. While the interest of using DNN to study partial differential equations is arising, here we adapt it to study kinetic models, in particular the linear transport model. Moreover, theoretical analysis on the convergence of neural network and its approximated solution towards analytic solution is shown. We demonstrate the accuracy and effectiveness of the proposed DNN method in numerical experiments.

Citation: Zheng Chen, Liu Liu, Lin Mu. Solving the linear transport equation by a deep neural network approach. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021070
References:
[1]

R. E. Alcouffe, A first Collision Source Method for Coupling Monte Carlo and Discrete Ordinates for Localized Source Problems, in Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics, Springer, 1985,352–366. Google Scholar

[2]

J.-F. BourgatP. Le Tallec and M. Tidriri, Coupling boltzmann and Navier–Stokes equations by friction, Journal of Computational Physics, 127 (1996), 227-245.  doi: 10.1006/jcph.1996.0172.  Google Scholar

[3] T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas, Cambridge University Press, 2003.  doi: 10.1017/CBO9780511755750.  Google Scholar
[4]

S. BrunnerE. Valeo and J. A. Krommes, Collisional delta-f scheme with evolving background for transport time scale simulations, Physics of Plasmas, 6 (1999), 4504-4521.  doi: 10.1063/1.873738.  Google Scholar

[5]

R. H. ByrdP. LuJ. Nocedal and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM Journal on Scientific Computing, 16 (1995), 1190-1208.  doi: 10.1137/0916069.  Google Scholar

[6]

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley, 1967.  Google Scholar

[7]

C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[8]

C. Cercignani, The Boltzmann equation in the whole space, in The Boltzmann Equation and Its Applications, Springer, 1988, 40–103. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[9]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[10] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge university press, 1970.   Google Scholar
[11]

Z. Chen and C. Hauck, Multiscale convergence properties for spectral approximations of a model kinetic equation, Mathematics of Computation, 88 (2019), 2257-2293.  doi: 10.1090/mcom/3399.  Google Scholar

[12]

Z. ChenL. Liu and L. Mu, Dg-imex stochastic galerkin schemes for linear transport equation with random inputs and diffusive scalings, Journal of Scientific Computing, 73 (2017), 566-592.  doi: 10.1007/s10915-017-0439-2.  Google Scholar

[13]

J. A. Coakley Jr. and P. Yang, Atmospheric Radiation: A Primer with Illustrative Solutions, John Wiley & Sons, 2014. Google Scholar

[14]

M. M. CrockattA. J. ChristliebC. K. Garrett and C. D. Hauck, An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction, Journal of Computational Physics, 346 (2017), 212-241.  doi: 10.1016/j.jcp.2017.06.017.  Google Scholar

[15]

G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signal Systems, 2 (1989), 303-314.  doi: 10.1007/BF02551274.  Google Scholar

[16]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology: Volume 1 Physical Origins and Classical Methods, Springer Science & Business Media, 2012.  Google Scholar

[17] B. Davison and J. B. Sykes, Neutron Transport Theory, Clarendon Press, 1957.   Google Scholar
[18]

V. P. DeCaria, C. D. Hauck and M. P. Laiu, Analysis of a new implicit solver for a semiconductor model, preprint, arXiv: 2009.05626, (2020). Google Scholar

[19]

P. Degond and S. Jin, A smooth transition model between kinetic and diffusion equations, SIAM Journal on Numerical Analysis, 42 (2005), 2671-2687.  doi: 10.1137/S0036142903430414.  Google Scholar

[20]

P. DegondS. Jin and L. Mieussens, A smooth transition model between kinetic and hydrodynamic equations, Journal of Computational Physics, 209 (2005), 665-694.  doi: 10.1016/j.jcp.2005.03.025.  Google Scholar

[21]

P. DegondJ.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects, Multiscale Modeling & Simulation, 5 (2006), 940-979.  doi: 10.1137/060651574.  Google Scholar

[22]

G. Dimarco and L. Pareschi, Hybrid multiscale methods ii. Kinetic equations, Multiscale Modeling & Simulation, 6 (2008), 1169-1197.  doi: 10.1137/070680916.  Google Scholar

[23]

G. Dimarco and L. Pareschi, Fluid solver independent hybrid methods for multiscale kinetic equations, SIAM Journal on Scientific Computing, 32 (2010), 603-634.  doi: 10.1137/080730585.  Google Scholar

[24]

I. M. GambaS. Jin and L. Liu, Micro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations, J. Comput. Phys., 382 (2019), 264-290.  doi: 10.1016/j.jcp.2019.01.018.  Google Scholar

[25]

C. K. Garrett and C. D. Hauck, A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport Theory and Statistical Physics, 42 (2013), 203-235.  doi: 10.1080/00411450.2014.910226.  Google Scholar

[26]

F. GolseS. Jin and C. D. Levermore, A domain decomposition analysis for a two-scale linear transport problem, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 869-892.  doi: 10.1051/m2an:2003059.  Google Scholar

[27]

C. Hauck and V. Heningburg, Filtered discrete ordinates equations for radiative transport, Journal of Scientific Computing, 80 (2019), 614-648.  doi: 10.1007/s10915-019-00950-1.  Google Scholar

[28]

C. Hauck and R. McClarren, Positive p_n closures, SIAM Journal on Scientific Computing, 32 (2010), 2603-2626.  doi: 10.1137/090764918.  Google Scholar

[29]

C. D. Hauck and R. G. McClarren, A collision-based hybrid method for time-dependent, linear, kinetic transport equations, Multiscale Modeling & Simulation, 11 (2013), 1197-1227.  doi: 10.1137/110846610.  Google Scholar

[30]

R. D. Hazeltine and F. L. Waelbroeck, The Framework of Plasma Physics, Westview, 2004. doi: 10.1201/9780429502804.  Google Scholar

[31]

V. Heningburg and C. D. Hauck, Hybrid solver for the radiative transport equation using finite volume and discontinuous galerkin, preprint, arXiv: 2002.02517, (2020). doi: 10.1137/19M1304520.  Google Scholar

[32]

H. J. Hwang, J. W. Jang, H. Jo and J. Y. Lee, Trend to equilibrium for the kinetic Fokker-Planck equation via the neural network approach, preprint, (2019). doi: 10.1016/j.jcp.2020.109665.  Google Scholar

[33]

J. JangF. LiJ.-M. Qiu and T. Xiong, High order asymptotic preserving dg-imex schemes for discrete-velocity kinetic equations in a diffusive scaling, Journal of Computational Physics, 281 (2015), 199-224.  doi: 10.1016/j.jcp.2014.10.025.  Google Scholar

[34]

S. Jin, Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations, SIAM Journal on Scientific Computing, 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.  Google Scholar

[35]

S. Jin, Asymptotic preserving (ap) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Mat. Univ. Parma, 3 (2012), 177-216.   Google Scholar

[36]

D. Kingma and J. Ba, Adam: A method for stochastic optimization, International Conference on Learning Representations, (2014). Google Scholar

[37]

A. Klar, Domain decomposition for kinetic problems with nonequilibrium states, Eur. J. Mech. B: Fluid, 15 (1996), 203-216.   Google Scholar

[38]

A. KlarH. Neunzert and J. Struckmeier, Transition from kinetic theory to macroscopic fluid equations: A problem for domain decomposition and a source for new algorithms, Transport Theory and Statistical Physics, 29 (2000), 93-106.  doi: 10.1080/00411450008205862.  Google Scholar

[39]

H. Kurt, M. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2, (1989), 359–366. Google Scholar

[40]

M. P. LaiuC. D. HauckR. G. McClarrenD. P. O'Leary and A. L. Tits, Positive filtered p _n moment closures for linear kinetic equations, SIAM Journal on Numerical Analysis, 54 (2016), 3214-3238.  doi: 10.1137/15M1052871.  Google Scholar

[41]

K. D. Lathrop, Ray effects in discrete ordinates equations, Nuclear Science and Engineering, 32 (1968), 357-369.  doi: 10.13182/NSE68-4.  Google Scholar

[42]

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport, John Wiley and Sons, Inc., New York, NY, 1984. Google Scholar

[43]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Communications in Mathematical Physics, 246 (2004), 133-179.  doi: 10.1007/s00220-003-1030-2.  Google Scholar

[44]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[45]

R. G. McClarren and C. D. Hauck, Robust and accurate filtered spherical harmonics expansions for radiative transfer, Journal of Computational Physics, 229 (2010), 5597-5614.  doi: 10.1016/j.jcp.2010.03.043.  Google Scholar

[46]

W. S. McCulluoch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys., 5 (1943), 115-133.  doi: 10.1007/BF02478259.  Google Scholar

[47]

A. Mezzacappa and O. Messer, Neutrino transport in core collapse supernovae, Journal of Computational and Applied Mathematics, 109 (1999), 281-319.   Google Scholar

[48]

D. Mihalas and B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics, Courier Corporation, 1999.  Google Scholar

[49]

S. Parker and W. Lee, A fully nonlinear characteristic method for gyrokinetic simulation, Physics of Fluids B: Plasma Physics, 5 (1993), 77-86.   Google Scholar

[50] A. Peraiah, An Introduction to Radiative Transfer: Methods and Applications in Astrophysics, Cambridge University Press, 2002.   Google Scholar
[51] G. C. Pomraning, Radiation Hydrodynamics, Pergamon Press, New York, 1973.  doi: 10.2172/656708.  Google Scholar
[52]

M. RaissiP. 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.  doi: 10.1016/j.jcp.2018.10.045.  Google Scholar

[53]

M. RaissiP. 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.  doi: 10.1016/j.jcp.2018.10.045.  Google Scholar

[54]

S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer Science & Business Media, 2012. doi: 10.1007/978-3-7091-8752-4.  Google Scholar

[55] K. StamnesG. E. Thomas and J. J. Stamnes, Radiative Transfer in the Atmosphere and Ocean, Cambridge University Press, 2017.  doi: 10.1017/9781316148549.  Google Scholar
[56]

A. Tartakovsky, C. Marrero, D. Tartakovsky and D. Barajas-Solano, Learning parameters and constitutive relationships with physics informed deep neural networks, preprint, arXiv: 1808.03398, (2018). doi: 10.1016/j.jcp.2019.06.041.  Google Scholar

[57] W. ZdunkowskiT. Trautmann and A. Bott, Radiation in the Atmosphere: A Course in Theoretical Meteorology, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511535796.  Google Scholar
[58]

Y. ZhuN. ZabarasP.-S. Koutsourelakis and P. Perdikaris, Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data, J. Comput. Phys., 394 (2019), 56-81.  doi: 10.1016/j.jcp.2019.05.024.  Google Scholar

show all references

References:
[1]

R. E. Alcouffe, A first Collision Source Method for Coupling Monte Carlo and Discrete Ordinates for Localized Source Problems, in Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics, Springer, 1985,352–366. Google Scholar

[2]

J.-F. BourgatP. Le Tallec and M. Tidriri, Coupling boltzmann and Navier–Stokes equations by friction, Journal of Computational Physics, 127 (1996), 227-245.  doi: 10.1006/jcph.1996.0172.  Google Scholar

[3] T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas, Cambridge University Press, 2003.  doi: 10.1017/CBO9780511755750.  Google Scholar
[4]

S. BrunnerE. Valeo and J. A. Krommes, Collisional delta-f scheme with evolving background for transport time scale simulations, Physics of Plasmas, 6 (1999), 4504-4521.  doi: 10.1063/1.873738.  Google Scholar

[5]

R. H. ByrdP. LuJ. Nocedal and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM Journal on Scientific Computing, 16 (1995), 1190-1208.  doi: 10.1137/0916069.  Google Scholar

[6]

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley, 1967.  Google Scholar

[7]

C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[8]

C. Cercignani, The Boltzmann equation in the whole space, in The Boltzmann Equation and Its Applications, Springer, 1988, 40–103. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[9]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[10] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge university press, 1970.   Google Scholar
[11]

Z. Chen and C. Hauck, Multiscale convergence properties for spectral approximations of a model kinetic equation, Mathematics of Computation, 88 (2019), 2257-2293.  doi: 10.1090/mcom/3399.  Google Scholar

[12]

Z. ChenL. Liu and L. Mu, Dg-imex stochastic galerkin schemes for linear transport equation with random inputs and diffusive scalings, Journal of Scientific Computing, 73 (2017), 566-592.  doi: 10.1007/s10915-017-0439-2.  Google Scholar

[13]

J. A. Coakley Jr. and P. Yang, Atmospheric Radiation: A Primer with Illustrative Solutions, John Wiley & Sons, 2014. Google Scholar

[14]

M. M. CrockattA. J. ChristliebC. K. Garrett and C. D. Hauck, An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction, Journal of Computational Physics, 346 (2017), 212-241.  doi: 10.1016/j.jcp.2017.06.017.  Google Scholar

[15]

G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signal Systems, 2 (1989), 303-314.  doi: 10.1007/BF02551274.  Google Scholar

[16]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology: Volume 1 Physical Origins and Classical Methods, Springer Science & Business Media, 2012.  Google Scholar

[17] B. Davison and J. B. Sykes, Neutron Transport Theory, Clarendon Press, 1957.   Google Scholar
[18]

V. P. DeCaria, C. D. Hauck and M. P. Laiu, Analysis of a new implicit solver for a semiconductor model, preprint, arXiv: 2009.05626, (2020). Google Scholar

[19]

P. Degond and S. Jin, A smooth transition model between kinetic and diffusion equations, SIAM Journal on Numerical Analysis, 42 (2005), 2671-2687.  doi: 10.1137/S0036142903430414.  Google Scholar

[20]

P. DegondS. Jin and L. Mieussens, A smooth transition model between kinetic and hydrodynamic equations, Journal of Computational Physics, 209 (2005), 665-694.  doi: 10.1016/j.jcp.2005.03.025.  Google Scholar

[21]

P. DegondJ.-G. Liu and L. Mieussens, Macroscopic fluid models with localized kinetic upscaling effects, Multiscale Modeling & Simulation, 5 (2006), 940-979.  doi: 10.1137/060651574.  Google Scholar

[22]

G. Dimarco and L. Pareschi, Hybrid multiscale methods ii. Kinetic equations, Multiscale Modeling & Simulation, 6 (2008), 1169-1197.  doi: 10.1137/070680916.  Google Scholar

[23]

G. Dimarco and L. Pareschi, Fluid solver independent hybrid methods for multiscale kinetic equations, SIAM Journal on Scientific Computing, 32 (2010), 603-634.  doi: 10.1137/080730585.  Google Scholar

[24]

I. M. GambaS. Jin and L. Liu, Micro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations, J. Comput. Phys., 382 (2019), 264-290.  doi: 10.1016/j.jcp.2019.01.018.  Google Scholar

[25]

C. K. Garrett and C. D. Hauck, A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport Theory and Statistical Physics, 42 (2013), 203-235.  doi: 10.1080/00411450.2014.910226.  Google Scholar

[26]

F. GolseS. Jin and C. D. Levermore, A domain decomposition analysis for a two-scale linear transport problem, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 869-892.  doi: 10.1051/m2an:2003059.  Google Scholar

[27]

C. Hauck and V. Heningburg, Filtered discrete ordinates equations for radiative transport, Journal of Scientific Computing, 80 (2019), 614-648.  doi: 10.1007/s10915-019-00950-1.  Google Scholar

[28]

C. Hauck and R. McClarren, Positive p_n closures, SIAM Journal on Scientific Computing, 32 (2010), 2603-2626.  doi: 10.1137/090764918.  Google Scholar

[29]

C. D. Hauck and R. G. McClarren, A collision-based hybrid method for time-dependent, linear, kinetic transport equations, Multiscale Modeling & Simulation, 11 (2013), 1197-1227.  doi: 10.1137/110846610.  Google Scholar

[30]

R. D. Hazeltine and F. L. Waelbroeck, The Framework of Plasma Physics, Westview, 2004. doi: 10.1201/9780429502804.  Google Scholar

[31]

V. Heningburg and C. D. Hauck, Hybrid solver for the radiative transport equation using finite volume and discontinuous galerkin, preprint, arXiv: 2002.02517, (2020). doi: 10.1137/19M1304520.  Google Scholar

[32]

H. J. Hwang, J. W. Jang, H. Jo and J. Y. Lee, Trend to equilibrium for the kinetic Fokker-Planck equation via the neural network approach, preprint, (2019). doi: 10.1016/j.jcp.2020.109665.  Google Scholar

[33]

J. JangF. LiJ.-M. Qiu and T. Xiong, High order asymptotic preserving dg-imex schemes for discrete-velocity kinetic equations in a diffusive scaling, Journal of Computational Physics, 281 (2015), 199-224.  doi: 10.1016/j.jcp.2014.10.025.  Google Scholar

[34]

S. Jin, Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations, SIAM Journal on Scientific Computing, 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.  Google Scholar

[35]

S. Jin, Asymptotic preserving (ap) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Mat. Univ. Parma, 3 (2012), 177-216.   Google Scholar

[36]

D. Kingma and J. Ba, Adam: A method for stochastic optimization, International Conference on Learning Representations, (2014). Google Scholar

[37]

A. Klar, Domain decomposition for kinetic problems with nonequilibrium states, Eur. J. Mech. B: Fluid, 15 (1996), 203-216.   Google Scholar

[38]

A. KlarH. Neunzert and J. Struckmeier, Transition from kinetic theory to macroscopic fluid equations: A problem for domain decomposition and a source for new algorithms, Transport Theory and Statistical Physics, 29 (2000), 93-106.  doi: 10.1080/00411450008205862.  Google Scholar

[39]

H. Kurt, M. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2, (1989), 359–366. Google Scholar

[40]

M. P. LaiuC. D. HauckR. G. McClarrenD. P. O'Leary and A. L. Tits, Positive filtered p _n moment closures for linear kinetic equations, SIAM Journal on Numerical Analysis, 54 (2016), 3214-3238.  doi: 10.1137/15M1052871.  Google Scholar

[41]

K. D. Lathrop, Ray effects in discrete ordinates equations, Nuclear Science and Engineering, 32 (1968), 357-369.  doi: 10.13182/NSE68-4.  Google Scholar

[42]

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport, John Wiley and Sons, Inc., New York, NY, 1984. Google Scholar

[43]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Communications in Mathematical Physics, 246 (2004), 133-179.  doi: 10.1007/s00220-003-1030-2.  Google Scholar

[44]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[45]

R. G. McClarren and C. D. Hauck, Robust and accurate filtered spherical harmonics expansions for radiative transfer, Journal of Computational Physics, 229 (2010), 5597-5614.  doi: 10.1016/j.jcp.2010.03.043.  Google Scholar

[46]

W. S. McCulluoch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys., 5 (1943), 115-133.  doi: 10.1007/BF02478259.  Google Scholar

[47]

A. Mezzacappa and O. Messer, Neutrino transport in core collapse supernovae, Journal of Computational and Applied Mathematics, 109 (1999), 281-319.   Google Scholar

[48]

D. Mihalas and B. Weibel-Mihalas, Foundations of Radiation Hydrodynamics, Courier Corporation, 1999.  Google Scholar

[49]

S. Parker and W. Lee, A fully nonlinear characteristic method for gyrokinetic simulation, Physics of Fluids B: Plasma Physics, 5 (1993), 77-86.   Google Scholar

[50] A. Peraiah, An Introduction to Radiative Transfer: Methods and Applications in Astrophysics, Cambridge University Press, 2002.   Google Scholar
[51] G. C. Pomraning, Radiation Hydrodynamics, Pergamon Press, New York, 1973.  doi: 10.2172/656708.  Google Scholar
[52]

M. RaissiP. 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.  doi: 10.1016/j.jcp.2018.10.045.  Google Scholar

[53]

M. RaissiP. 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.  doi: 10.1016/j.jcp.2018.10.045.  Google Scholar

[54]

S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer Science & Business Media, 2012. doi: 10.1007/978-3-7091-8752-4.  Google Scholar

[55] K. StamnesG. E. Thomas and J. J. Stamnes, Radiative Transfer in the Atmosphere and Ocean, Cambridge University Press, 2017.  doi: 10.1017/9781316148549.  Google Scholar
[56]

A. Tartakovsky, C. Marrero, D. Tartakovsky and D. Barajas-Solano, Learning parameters and constitutive relationships with physics informed deep neural networks, preprint, arXiv: 1808.03398, (2018). doi: 10.1016/j.jcp.2019.06.041.  Google Scholar

[57] W. ZdunkowskiT. Trautmann and A. Bott, Radiation in the Atmosphere: A Course in Theoretical Meteorology, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511535796.  Google Scholar
[58]

Y. ZhuN. ZabarasP.-S. Koutsourelakis and P. Perdikaris, Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data, J. Comput. Phys., 394 (2019), 56-81.  doi: 10.1016/j.jcp.2019.05.024.  Google Scholar

Figure 1.  Structure of the deep neural network
Figure 2.  Example 5.2.1 with $ \sigma_s = 0.0 $ and $ \sigma_a = 1.0 $: (a). exact solution $ \rho $; (b). DNN approximation $ \rho_h $
Figure 3.  Example 5.2.1 with $ \sigma_s = 0.0,\sigma_a = 1.0 $: DNN approximation to $ \psi $ at time = 1.0
Figure 4.  Example 5.2.1: Plot of DNN solutions for $ \rho_h $: (a) $ \sigma_s = 1 $ and $ \sigma_a = 0 $; (b) $ \sigma_s = 1.0 $ and $ \sigma_a = 1.0 $
Figure 5.  Example 5.2.1 with $ \sigma_s = 1.0,\sigma_a = 0.0 $: DNN approximation to $ \psi $ at time = 1.0
Figure 6.  Example 5.2.1 with $ \sigma_s = 1.0,\sigma_a = 1.0 $: DNN approximation to $ \psi $ at time = 1.0
Figure 7.  Example 5.2.2: Plot of DNN approximation to $ \psi $ for $ \sigma_s = 1.0 $ and $ \sigma_a = 0.0 $ at different time
Figure 9.  Example 5.2.2: Plots of DNN solutions $ \rho_h $ for $ k = 100 $ and (a) $ \sigma_s = 1.0 $, $ \sigma_a = 0.0 $; (b)$ \sigma_s = 1.0 $, $ \sigma_a = 1.0 $
Figure 8.  Example 5.2.2: Plot of DNN approximation to $ \psi $ for $ \sigma_s = 1.0 $ and $ \sigma_a = 1.0 $ at different time
Figure 10.  Example 5.2.2: Plots of angular average of discrete ordinate $ S_{100} $ solutions for $ k = 100 $: (a) $ \sigma_s = 1 $, $ \sigma_a = 0 $; (b) $ \sigma_s = 1 $, $ \sigma_a = 1 $
Figure 11.  Example 5.2.3: The illustration of the boundary condition
Figure 12.  Example 5.2.3: Case (1) $ \sigma_s = 1.0 $, $ \sigma_a = 9.0 $: (a). solution of $ \rho_h $ at different time; (b). 2-dimensional plot of DNN approximation to $ \psi $ on the $ x-\mu $ plane at $ t = 10 $
Figure 13.  Example 5.2.3: Case (2) $ \sigma_s = 5.0 $, $ \sigma_a = 5.0 $: (a). solution of $ \rho_h $ at different time; (b). 2-dimensional plot of DNN approximation to $ \psi $ on the $ x-\mu $ plane at $ t = 10 $
Figure 14.  Example 5.2.3: Case (3) $ \sigma_s = 9.0 $, $ \sigma_a = 1.0 $: (a). solution of $ \rho_h $ at different time; (b). 2-dimensional plot of DNN approximation to $ \psi $ on the $ x-\mu $ plane at $ t = 10 $
Figure 15.  Example 5.2.3: Plots of DNN solutions of $ \rho_h $ at time = 10.0
Table 1.  Example 5.2.1: Relative Errors in DNN approximations to $ \psi $
time $ \sigma_s = 0.0,\sigma_a = 1.0 $ $ \sigma_s = 1.0,\sigma_a = 0.0 $ $ \sigma_s = 1.0,\sigma_a = 1.0 $
0.0 2.3329e-4 2.7779e-4 2.8335e-4
0.2 1.8372e-4 4.4767e-2 2.1264e-2
0.4 1.5604e-4 4.1943e-2 2.7555e-2
0.6 1.3525e-4 4.1276e-2 2.6886e-2
0.8 1.1590e-4 4.7595e-2 2.8699e-2
1.0 1.1543e-4 4.5483e-2 2.1431e-2
time $ \sigma_s = 0.0,\sigma_a = 1.0 $ $ \sigma_s = 1.0,\sigma_a = 0.0 $ $ \sigma_s = 1.0,\sigma_a = 1.0 $
0.0 2.3329e-4 2.7779e-4 2.8335e-4
0.2 1.8372e-4 4.4767e-2 2.1264e-2
0.4 1.5604e-4 4.1943e-2 2.7555e-2
0.6 1.3525e-4 4.1276e-2 2.6886e-2
0.8 1.1590e-4 4.7595e-2 2.8699e-2
1.0 1.1543e-4 4.5483e-2 2.1431e-2
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