• Previous Article
    Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise
  • DCDS-S Home
  • This Issue
  • Next Article
    A dictionary learning algorithm for compression and reconstruction of streaming data in preset order
April  2022, 15(4): 669-686. doi: 10.3934/dcdss.2021070

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 Published  April 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (4) : 669-686. 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.

[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.

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

[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.

[6]

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

[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.

[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.

[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.

[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. 
[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.

[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.

[13]

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

[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.

[15]

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

[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.

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[30]

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

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[37]

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

[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.

[39]

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

[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.

[41]

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

[42]

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

[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.

[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.

[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.

[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.

[47]

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

[48]

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

[49]

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

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

[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.

[54]

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

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

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

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.

[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.

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

[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.

[6]

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

[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.

[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.

[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.

[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. 
[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.

[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.

[13]

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

[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.

[15]

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

[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.

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[30]

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

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[37]

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

[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.

[39]

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

[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.

[41]

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

[42]

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

[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.

[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.

[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.

[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.

[47]

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

[48]

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

[49]

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

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

[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.

[54]

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

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

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

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
[1]

Hyeontae Jo, Hwijae Son, Hyung Ju Hwang, Eun Heui Kim. Deep neural network approach to forward-inverse problems. Networks and Heterogeneous Media, 2020, 15 (2) : 247-259. doi: 10.3934/nhm.2020011

[2]

Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283

[3]

Yuantian Xia, Juxiang Zhou, Tianwei Xu, Wei Gao. An improved deep convolutional neural network model with kernel loss function in image classification. Mathematical Foundations of Computing, 2020, 3 (1) : 51-64. doi: 10.3934/mfc.2020005

[4]

Boris Andreianov, Nicolas Seguin. Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1939-1964. doi: 10.3934/dcds.2012.32.1939

[5]

Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks and Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143

[6]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[7]

Leong-Kwan Li, Sally Shao. Convergence analysis of the weighted state space search algorithm for recurrent neural networks. Numerical Algebra, Control and Optimization, 2014, 4 (3) : 193-207. doi: 10.3934/naco.2014.4.193

[8]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2021, 8 (2) : 131-152. doi: 10.3934/jcd.2021006

[9]

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457

[10]

Kuo-Shou Chiu. Periodicity and stability analysis of impulsive neural network models with generalized piecewise constant delays. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 659-689. doi: 10.3934/dcdsb.2021060

[11]

Ling Zhang, Xiaoqi Sun. Stability analysis of time-varying delay neural network for convex quadratic programming with equality constraints and inequality constraints. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022035

[12]

G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure and Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509

[13]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems and Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077

[14]

Yong Duan, Jian-Guo Liu. Convergence analysis of the vortex blob method for the $b$-equation. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1995-2011. doi: 10.3934/dcds.2014.34.1995

[15]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[16]

Roman Romanov. Estimates of solutions of linear neutron transport equation at large time and spectral singularities. Kinetic and Related Models, 2012, 5 (1) : 113-128. doi: 10.3934/krm.2012.5.113

[17]

Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic and Related Models, 2014, 7 (1) : 79-108. doi: 10.3934/krm.2014.7.79

[18]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Stability of the dynamics of an asymmetric neural network. Communications on Pure and Applied Analysis, 2009, 8 (2) : 655-671. doi: 10.3934/cpaa.2009.8.655

[19]

Jiequn Han, Jihao Long. Convergence of the deep BSDE method for coupled FBSDEs. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 5-. doi: 10.1186/s41546-020-00047-w

[20]

Jiequn Han, Ruimeng Hu, Jihao Long. Convergence of deep fictitious play for stochastic differential games. Frontiers of Mathematical Finance, 2022, 1 (2) : 287-319. doi: 10.3934/fmf.2021011

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (335)
  • HTML views (506)
  • Cited by (0)

Other articles
by authors

[Back to Top]