Structure-preserving neural networks for the regularized entropy-based closure of a linear, kinetic, radiative transport equation

Schotthöfer, Steffen; Laiu, M. Paul; Frank, Martin; Hauck, Cory D.

doi:10.1016/j.jcp.2025.113967

Structure-preserving neural networks for the regularized entropy-based closure of a linear, kinetic, radiative transport equation

Journal Article · Wed Mar 26 00:00:00 EDT 2025 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2025.113967· OSTI ID:2586969

^[1]; Laiu, M. Paul ^[1]; Frank, Martin ^[2]; Hauck, Cory D. ^[3]

Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Karlsruhe Inst. of Technology (KIT) (Germany)
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States); Univ. of Tennessee, Knoxville, TN (United States)

The main challenge of large-scale numerical simulation of radiation transport is the high memory and computation time requirements of discretization methods for kinetic equations. In this work, we derive and investigate a neural network-based approximation to the entropy-based closure method to accurately compute the solution of the multi-dimensional moment system with a low memory footprint and competitive computational time. We extend methods developed for the standard entropy-based closure to the regularized entropy-based closures. The main idea is to interpret structure-preserving neural network approximations of the regularized entropy-based closure as a two-stage approximation to the original entropy-based closure. We conduct a numerical analysis of this approximation and investigate optimal parameter choices. Our numerical experiments demonstrate that the method has a much lower memory footprint than traditional methods with competitive computation times and simulation accuracy. The code and all trained networks are provided on GitHub.

View Accepted Manuscript (DOE)

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 2586969

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 533; ISSN 0021-9991

Publisher:: Elsevier BVCopyright Statement

Country of Publication:: United States

Language:: English

References (30)

Spherical Harmonics and Approximations on the Unit Sphere: An Introduction Atkinson, Kendall; Han, Weimin https://doi.org/10.1007/978-3-642-25983-8	book	January 2012
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
Positivity Limiters for Filtered Spectral Approximations of Linear Kinetic Transport Equations Laiu, M. Paul; Hauck, Cory D. Journal of Scientific Computing, Vol. 78, Issue 2 https://doi.org/10.1007/s10915-018-0790-y	journal	August 2018
Filtered Discrete Ordinates Equations for Radiative Transport Hauck, Cory; Heningburg, Vincent Journal of Scientific Computing, Vol. 80, Issue 1 https://doi.org/10.1007/s10915-019-00950-1	journal	May 2019
Machine Learning Moment Closure Models for the Radiative Transfer Equation III: Enforcing Hyperbolicity and Physical Characteristic Speeds Huang, Juntao; Cheng, Yingda; Christlieb, Andrew J. Journal of Scientific Computing, Vol. 94, Issue 1 https://doi.org/10.1007/s10915-022-02056-7	journal	November 2022
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
Optimization and large scale computation of an entropy-based moment closure Kristopher Garrett, C.; Hauck, Cory; Hill, Judith Journal of Computational Physics, Vol. 302 https://doi.org/10.1016/j.jcp.2015.09.008	journal	December 2015
Improvements to a class of hybrid methods for radiation transport: Nyström reconstruction and defect correction methods Crockatt, Michael M.; Christlieb, Andrew J.; Hauck, Cory D. Journal of Computational Physics, Vol. 422 https://doi.org/10.1016/j.jcp.2020.109765	journal	December 2020
Machine learning moment closure models for the radiative transfer equation I: Directly learning a gradient based closure Huang, Juntao; Cheng, Yingda; Christlieb, Andrew J. Journal of Computational Physics, Vol. 453 https://doi.org/10.1016/j.jcp.2022.110941	journal	March 2022
Predicting continuum breakdown with deep neural networks Xiao, Tianbai; Schotthöfer, Steffen; Frank, Martin Journal of Computational Physics, Vol. 489 https://doi.org/10.1016/j.jcp.2023.112278	journal	September 2023
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
Systems of Conservation Equations with a Convex Extension Friedrichs, K. O.; Lax, P. D. Proceedings of the National Academy of Sciences, Vol. 68, Issue 8 https://doi.org/10.1073/pnas.68.8.1686	journal	August 1971
Ray Effect Mitigation for the Discrete Ordinates Method Using Artificial Scattering Frank, Martin; Kusch, Jonas; Camminady, Thomas Nuclear Science and Engineering, Vol. 194, Issue 11 https://doi.org/10.1080/00295639.2020.1730665	journal	March 2020
A Comparison of Moment Closures for Linear Kinetic Transport Equations: The Line Source Benchmark Garrett, C. Kristopher; Hauck, Cory D. Transport Theory and Statistical Physics, Vol. 42, Issue 6-7 https://doi.org/10.1080/00411450.2014.910226	journal	September 2013
Entropy-based moment closures for kinetic equations Levermore, C. David Transport Theory and Statistical Physics, Vol. 26, Issue 4-5 https://doi.org/10.1080/00411459708017931	journal	January 1997
Dual Methods in Entropy Maximization. Application to Some Problems in Crystallography Decarreau, Andrée; Hilhorst, Danielle; Lemaréchal, Claude SIAM Journal on Optimization, Vol. 2, Issue 2 https://doi.org/10.1137/0802010	journal	May 1992
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
Foundations of Radiation Hydrodynamics (Dimitri Mihalas and Barbara Weibel Mihalas) Chahine, Moustafa T. SIAM Review, Vol. 29, Issue 4 https://doi.org/10.1137/1029132	journal	December 1987
A Collision-Based Hybrid Method for Time-Dependent, Linear, Kinetic Transport Equations Hauck, Cory D.; McClarren, Ryan G. Multiscale Modeling & Simulation, Vol. 11, Issue 4 https://doi.org/10.1137/110846610	journal	January 2013
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
Positive Filtered P$_N$ Moment Closures for Linear Kinetic Equations Laiu, M. Paul; Hauck, Cory D.; McClarren, Ryan G. SIAM Journal on Numerical Analysis, Vol. 54, Issue 6 https://doi.org/10.1137/15M1052871	journal	January 2016
A Regularized Entropy-Based Moment Method for Kinetic Equations Alldredge, Graham W.; Frank, Martin; Hauck, Cory D. SIAM Journal on Applied Mathematics, Vol. 79, Issue 5 https://doi.org/10.1137/18M1181201	journal	January 2019
A Positive Asymptotic-Preserving Scheme for Linear Kinetic Transport Equations Laiu, M. Paul; Frank, Martin; Hauck, Cory D. SIAM Journal on Scientific Computing, Vol. 41, Issue 3 https://doi.org/10.1137/18M1196297	journal	January 2019
Machine Learning Moment Closure Models for the Radiative Transfer Equation II: Enforcing Global Hyperbolicity in Gradient-Based Closures Huang, Juntao; Cheng, Yingda; Christlieb, Andrew J. Multiscale Modeling & Simulation, Vol. 21, Issue 2 https://doi.org/10.1137/21M1423956	journal	April 2023
Ray Effects in Discrete Ordinates Equations Lathrop, K. D. Nuclear Science and Engineering, Vol. 32, Issue 3 https://doi.org/10.13182/NSE68-4	journal	June 1968
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
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
Data-driven, structure-preserving approximations to entropy-based moment closures for kinetic equations Porteous, William A.; Laiu, Ming Tse P.; Hauck, Cory D. Communications in Mathematical Sciences, Vol. 21, Issue 4 https://doi.org/10.4310/CMS.2023.v21.n4.a1	journal	January 2023
On the convergence of the regularized entropy-based moment method for kinetic equations Alldredge, Graham W.; Frank, Martin; Giesselmann, Jan The SMAI Journal of computational mathematics, Vol. 9 https://doi.org/10.5802/smai-jcm.93	journal	January 2023

Similar Records

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

Journal Article · Fri Mar 24 00:00:00 EDT 2023 · Communications in Mathematical Sciences · OSTI ID:1999103

A Regularized Entropy-Based Moment Method for Kinetic Equations

Journal Article · Thu Sep 05 00:00:00 EDT 2019 · SIAM Journal of Applied Mathematics · OSTI ID:1845814

Efficient algorithms and implementations of entropy-based moment closures for rarefied gases

Journal Article · Sat Jul 01 00:00:00 EDT 2017 · Journal of Computational Physics · OSTI ID:22622300

Related Subjects

entropy closure
kinetic theory
moment methods
neural networks
regularized optimization

Structure-preserving neural networks for the regularized entropy-based closure of a linear, kinetic, radiative transport equation

Citation Formats

References (30)

Similar Records

Related Subjects