Machine learning prediction of discrete eigenvalues for the radiative transfer equation under anisotropic Henyey–Greenstein scattering
- 1Vocational School of Health Services, Iğdır University, Iğdır, 76000, Turkey; Corresponding author.
- 2Vocational School of Technical Sciences, Iğdır University, Iğdır, 76000, Turkey
Abstract
The discrete eigenvalues of the Radiative Transfer Equation (RTE) play a fundamental role in the analysis of radiation transport under anisotropic scattering conditions. For the Henyey–Greenstein phase function, these eigenvalues depend on the single-scattering albedo (ω) and the anisotropy parameter (g). Although the Legendre polynomial (P 9 ) approximation provides accurate reference solutions, each parameter combination requires a separate numerical solution.In this study, a multi-output machine learning framework is developed to predict the five dominant discrete eigenvalues of the RTE directly from the optical parameters. A reference dataset was generated using a Mathematica implementation of the P 9 method and used to train three regression models: Random Forest (RF), Gradient Boosting (GB), and Multi-Layer Perceptron (MLP). Model performance was assessed using standard statistical error metrics and comparisons with the reference P 9 solutions.All three models achieved high predictive accuracy (R 2 > 0.95), with the MLP model providing the best overall performance, particularly for the higher-order eigenvalues. Once trained, the surrogate models estimate the five dominant eigenvalues directly from the optical parameters without repeated execution of the P 9 solution procedure. These results demonstrate that conventional multi-output machine learning models provide an accurate and practical surrogate approach for repeated eigenvalue predictions within the investigated parameter domain.
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BibTeX
@article{koklu2026machine,
title = {{Machine learning prediction of discrete eigenvalues for the radiative transfer equation under anisotropic Henyey–Greenstein scattering}},
author = {Halide Köklü and Metehan Günde},
journal = {Physics Open},
year = {2026},
month = oct,
doi = {10.1016/j.physo.2026.100502},
url = {https://doi.org/10.1016/j.physo.2026.100502},
note = {CC BY},
}
RIS
TY - JOUR
TI - Machine learning prediction of discrete eigenvalues for the radiative transfer equation under anisotropic Henyey–Greenstein scattering
AU - Halide Köklü
AU - Metehan Günde
PY - 2026
DA - 2026/10/30
JO - Physics Open
DO - 10.1016/j.physo.2026.100502
UR - https://doi.org/10.1016/j.physo.2026.100502
AB - The discrete eigenvalues of the Radiative Transfer Equation (RTE) play a fundamental role in the analysis of radiation transport under anisotropic scattering conditions. For the Henyey–Greenstein phase function, these eigenvalues depend on the single-scattering albedo (ω) and the anisotropy parameter (g). Although the Legendre polynomial (P 9 ) approximation provides accurate reference solutions, each parameter combination requires a separate numerical solution.In this study, a multi-output machine learning framework is developed to predict the five dominant discrete eigenvalues of the RTE directly from the optical parameters. A reference dataset was generated using a Mathematica implementation of the P 9 method and used to train three regression models: Random Forest (RF), Gradient Boosting (GB), and Multi-Layer Perceptron (MLP). Model performance was assessed using standard statistical error metrics and comparisons with the reference P 9 solutions.All three models achieved high predictive accuracy (R 2 > 0.95), with the MLP model providing the best overall performance, particularly for the higher-order eigenvalues. Once trained, the surrogate models estimate the five dominant eigenvalues directly from the optical parameters without repeated execution of the P 9 solution procedure. These results demonstrate that conventional multi-output machine learning models provide an accurate and practical surrogate approach for repeated eigenvalue predictions within the investigated parameter domain.
SN - 2666-0326
ER -