Machine learning moment closure models for the radiative transfer equation I: Directly learning a gradient based closure

Huang, Juntao; Cheng, Yingda; Christlieb, Andrew J.; Roberts, Luke F.

doi:10.1016/j.jcp.2022.110941

Title: Machine learning moment closure models for the radiative transfer equation I: Directly learning a gradient based closure

Journal Article · 06 January 2022 · Journal of Computational Physics

DOI: https://doi.org/10.1016/j.jcp.2022.110941 · OSTI ID:1977277

^[1]; Cheng, Yingda ^[2]; Christlieb, Andrew J. ^[2];

^[2]

Michigan State Univ., East Lansing, MI (United States); OSTI
Michigan State Univ., East Lansing, MI (United States)

In this paper, we take a data-driven approach and apply machine learning to the moment closure problem for the radiative transfer equation in slab geometry. Instead of learning the unclosed high order moment, we propose to directly learn the gradient of the high order moment using neural networks. This new approach is consistent with the exact closure we derive for the free streaming limit and also provides a natural output normalization. Finally, a variety of benchmark tests, including the variable scattering problem, the Gaussian source problem with both periodic and reflecting boundaries, and the two-material problem, show both good accuracy and generalizability of our machine learning closure model.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Michigan State Univ., East Lansing, MI (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: SC0017955

OSTI ID:: 1977277

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 453; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (40)

Globally Hyperbolic Regularization of Grad's Moment System Cai, Zhenning; Fan, Yuwei; Li, Ruo Communications on Pure and Applied Mathematics, Vol. 67, Issue 3 https://doi.org/10.1002/cpa.21472	journal	July 2013
Efficient Implementation of Weighted ENO Schemes Jiang, Guang-Shan; Shu, Chi-Wang Journal of Computational Physics, Vol. 126, Issue 1 https://doi.org/10.1006/jcph.1996.0130	journal	June 1996
Moment closure hierarchies for kinetic theories Levermore, C. David Journal of Statistical Physics, Vol. 83, Issue 5-6 https://doi.org/10.1007/BF02179552	journal	June 1996
Efficient implementation of essentially non-oscillatory shock-capturing schemes Shu, Chi-Wang; Osher, Stanley Journal of Computational Physics, Vol. 77, Issue 2 https://doi.org/10.1016/0021-9991(88)90177-5	journal	August 1988
Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II larsen, Edward W.; Morel, J. E. Journal of Computational Physics, Vol. 83, Issue 1 https://doi.org/10.1016/0021-9991(89)90229-5	journal	July 1989
Relating Eddington factors to flux limiters Levermore, C. D. Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 31, Issue 2 https://doi.org/10.1016/0022-4073(84)90112-2	journal	February 1984
A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval Spurr, R. J. D.; Kurosu, T. P.; Chance, K. V. Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 68, Issue 6 https://doi.org/10.1016/S0022-4073(00)00055-8	journal	March 2001
Optical tomography using the time-independent equation of radiative transfer — Part 1: forward model Klose, Alexander D.; Netz, Uwe; Beuthan, Jürgen Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 72, Issue 5 https://doi.org/10.1016/S0022-4073(01)00150-9	journal	March 2002
Evaluation of quadrature schemes for the discrete ordinates method Koch, Rainer; Becker, Ralf Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 84, Issue 4 https://doi.org/10.1016/S0022-4073(03)00260-7	journal	April 2004
Robust and accurate filtered spherical harmonics expansions for radiative transfer McClarren, Ryan G.; Hauck, Cory D. Journal of Computational Physics, Vol. 229, Issue 16 https://doi.org/10.1016/j.jcp.2010.03.043	journal	August 2010
A new spherical harmonics scheme for multi-dimensional radiation transport I. Static matter configurations Radice, David; Abdikamalov, Ernazar; Rezzolla, Luciano Journal of Computational Physics, Vol. 242 https://doi.org/10.1016/j.jcp.2013.01.048	journal	June 2013
Adaptive change of basis in entropy-based moment closures for linear kinetic equations Alldredge, Graham W.; Hauck, Cory D.; OʼLeary, Dianne P. Journal of Computational Physics, Vol. 258 https://doi.org/10.1016/j.jcp.2013.10.049	journal	February 2014
A POD reduced order model for resolving angular direction in neutron/photon transport problems Buchan, A. G.; Calloo, A. A.; Goffin, M. G. Journal of Computational Physics, Vol. 296 https://doi.org/10.1016/j.jcp.2015.04.043	journal	September 2015
Implicit filtered P for high-energy density thermal radiation transport using discontinuous Galerkin finite elements Laboure, Vincent M.; McClarren, Ryan G.; Hauck, Cory D. Journal of Computational Physics, Vol. 321 https://doi.org/10.1016/j.jcp.2016.05.046	journal	September 2016
An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction Crockatt, Michael M.; Christlieb, Andrew J.; Garrett, C. Kristopher Journal of Computational Physics, Vol. 346 https://doi.org/10.1016/j.jcp.2017.06.017	journal	October 2017
Hybrid methods for radiation transport using diagonally implicit Runge–Kutta and space–time discontinuous Galerkin time integration Crockatt, Michael M.; Christlieb, Andrew J.; Garrett, C. Kristopher Journal of Computational Physics, Vol. 376 https://doi.org/10.1016/j.jcp.2018.09.041	journal	January 2019
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations Raissi, M.; Perdikaris, P.; Karniadakis, G. E. Journal of Computational Physics, Vol. 378 https://doi.org/10.1016/j.jcp.2018.10.045	journal	February 2019
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders Lee, Kookjin; Carlberg, Kevin T. Journal of Computational Physics https://doi.org/10.1016/j.jcp.2019.108973	journal	November 2019
A nonlinear moment model for radiative transfer equation in slab geometry Fan, Yuwei; Li, Ruo; Zheng, Lingchao Journal of Computational Physics, Vol. 404 https://doi.org/10.1016/j.jcp.2019.109128	journal	March 2020
Deep learning LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey Nature, Vol. 521, Issue 7553 https://doi.org/10.1038/nature14539	journal	May 2015
Machine learning surrogate models for Landau fluid closure Ma, Chenhao; Zhu, Ben; Xu, Xue-Qiao Physics of Plasmas, Vol. 27, Issue 4 https://doi.org/10.1063/1.5129158	journal	April 2020
Neural network representability of fully ionized plasma fluid model closures Maulik, Romit; Garland, Nathan A.; Burby, Joshua W. Physics of Plasmas, Vol. 27, Issue 7 https://doi.org/10.1063/5.0006457	journal	July 2020
Deep learning surrogate model for kinetic Landau-fluid closure with collision Wang, Libo; Xu, X. Q.; Zhu, Ben AIP Advances, Vol. 10, Issue 7 https://doi.org/10.1063/5.0010917	journal	July 2020
Discovering governing equations from data by sparse identification of nonlinear dynamical systems Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan Proceedings of the National Academy of Sciences, Vol. 113, Issue 15 https://doi.org/10.1073/pnas.1517384113	journal	March 2016
Solving high-dimensional partial differential equations using deep learning Han, Jiequn; Jentzen, Arnulf; E., Weinan Proceedings of the National Academy of Sciences, Vol. 115, Issue 34 https://doi.org/10.1073/pnas.1718942115	journal	August 2018
Uniformly accurate machine learning-based hydrodynamic models for kinetic equations Han, Jiequn; Ma, Chao; Ma, Zheng Proceedings of the National Academy of Sciences, Vol. 116, Issue 44 https://doi.org/10.1073/pnas.1909854116	journal	October 2019
On the Radiative Equilibrium of a Stellar Atmosphere. Chandrasekhar, S. The Astrophysical Journal, Vol. 99 https://doi.org/10.1086/144606	journal	March 1944
Analytic closures for M1 neutrino transport Murchikova, E. M.; Abdikamalov, E.; Urbatsch, T. Monthly Notices of the Royal Astronomical Society, Vol. 469, Issue 2 https://doi.org/10.1093/mnras/stx986	journal	April 2017
Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics Zhang, Linfeng; Han, Jiequn; Wang, Han Physical Review Letters, Vol. 120, Issue 14 https://doi.org/10.1103/PhysRevLett.120.143001	journal	April 2018
Positive $P_N$ Closures Hauck, Cory; McClarren, Ryan SIAM Journal on Scientific Computing, Vol. 32, Issue 5 https://doi.org/10.1137/090764918	journal	January 2010
High-Order Entropy-Based Closures for Linear Transport in Slab Geometry II: A Computational Study of the Optimization Problem Alldredge, Graham W.; Hauck, Cory D.; Tits, André L. SIAM Journal on Scientific Computing, Vol. 34, Issue 4 https://doi.org/10.1137/11084772X	journal	January 2012
A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations Guo, Wei; Cheng, Yingda SIAM Journal on Scientific Computing, Vol. 38, Issue 6 https://doi.org/10.1137/16M1060017	journal	January 2016
An Adaptive Multiresolution Discontinuous Galerkin Method for Time-Dependent Transport Equations in Multidimensions Guo, Wei; Cheng, Yingda SIAM Journal on Scientific Computing, Vol. 39, Issue 6 https://doi.org/10.1137/16M1083190	journal	January 2017
Deep Learning: An Introduction for Applied Mathematicians Higham, Catherine F.; Higham, Desmond J. SIAM Review, Vol. 61, Issue 3 https://doi.org/10.1137/18M1165748	journal	January 2019
A Nonlinear Hyperbolic Model for Radiative Transfer Equation in Slab Geometry Fan, Yuwei; Li, Ruo; Zheng, Lingchao SIAM Journal on Applied Mathematics, Vol. 80, Issue 6 https://doi.org/10.1137/19M126774X	journal	January 2020
Conservation-dissipation formalism of irreversible thermodynamics Zhu, Yi; Hong, Liu; Yang, Zaibao Journal of Non-Equilibrium Thermodynamics, Vol. 40, Issue 2 https://doi.org/10.1515/jnet-2014-0037	journal	January 2015
Learning Thermodynamically Stable and Galilean Invariant Partial Differential Equations for Non-Equilibrium Flows Huang, Juntao; Ma, Zhiting; Zhou, Yizhou Journal of Non-Equilibrium Thermodynamics, Vol. 46, Issue 4 https://doi.org/10.1515/jnet-2021-0008	journal	May 2021
Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry Alldredge, Graham W.; Li, Ruo; Li, Weiming Kinetic and Related Models, Vol. 9, Issue 2 https://doi.org/10.3934/krm.2016.9.237	journal	March 2016
Numerical Study of Partially Conservative Moment Equations in Kinetic Theory Koellermeier, Julian; Torrilhon, Manuel Communications in Computational Physics, Vol. 21, Issue 4 https://doi.org/10.4208/cicp.OA-2016-0053	journal	March 2017
High-order entropy-based closures for linear transport in slab geometry Hauck, Cory D. Communications in Mathematical Sciences, Vol. 9, Issue 1 https://doi.org/10.4310/CMS.2011.v9.n1.a9	journal	January 2011

Similar Records

Machine Learning Moment Closure Models for the Radiative Transfer Equation III: Enforcing Hyperbolicity and Physical Characteristic Speeds

Journal Article · 2022 · Journal of Scientific Computing · OSTI ID:2282528

Huang, Juntao; Cheng, Yingda; Christlieb, Andrew J.; +1 more

Machine learning for reparameterization of multi-scale closures

Journal Article · 2025 · Machine Learning: Science and Technology · OSTI ID:2574711

Egan, Hilary; Crowley, Meagan; Sitaraman, Hariswaran; +2 more

Data-driven, structure-preserving approximations to entropy-based moment closures for kinetic equations

Journal Article · 2023 · Communications in Mathematical Sciences · OSTI ID:1999103

Porteous, William A.; Laiu, Ming Tse P.; Hauck, Cory D.

Related Subjects

Computer science
Machine learning
Moment closure
Neural network
Physics
Radiative transfer equation
Slab geometry

Title: Machine learning moment closure models for the radiative transfer equation I: Directly learning a gradient based closure

Citation Formats

References (40)

Similar Records

Related Subjects