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Deep Learning Application for Milne problem with linear anisotropic scattering

Yıl 2025, Erken Görünüm, 1 - 1

R. Gökhan Türeci

https://doi.org/10.35378/gujs.1521834

Öz

Using deep learning algorithms, which is a sub-branch of artificial intelligence, in this study, a deep learning model is developed according to both the number of secondary neutrons and the linear anisotropic scattering coefficient. These are independent variables that the dependent variable is the extrapolation distance. The training data set was calculated with HN method. ANN algorithm was written by TensorFlow and Keras which are the modules in Python programming language. The performance of the deep learning model for this problem has high performance so that the predicted new data which doesn’t be in the training data set is reliable according to the success of the model.

Anahtar Kelimeler

Deep Learning Milne problem Linear anisotropic scattering Artificial neural networks

Kaynakça

  • [1] Placzek, G., and W. Seidel., “Milne's Problem in Transport Theory”, Physical Review, 72: 550-555, (1947).
  • [2] Noble, B., The Wiener-Hopf Technique, Pergamon Press, Oxford, (1958).
  • [3] Case, K.M. and Zweifel, P.F., Linear Transport Theory, Massachusetts: Addition-Wesley, (1967).
  • [4] McCormick, N.J., “One-Speed Neutron Transport Problems in Plane Geometry”, PhD thesis, The University of Michigan, (1964).
  • [5] McCormick N. J., and Kuščer, I., “Half‐Space Neutron Transport with Linearly Anisotropic Scattering”, Journal of Mathematical Physics, 6 (12): 1939-1945, (1964).
  • [6] Kavenoky, A., “The CN Method of Solving the Transport Equation: Application to Plane Geometry”, Nuclear Science and Engineering, 65, 209, (1978).
  • [7] Sahni, D.C., and Kumar, V., “Numerical solution of singular integral equations of neutron transport problems”, Transport Theory and Statistical Physics, 16(7): 959-978, (1987).
  • [8] Atalay, M. A., “Milne problem for linearly anisotropic scattering and a specularly reflecting boundary”, Annals of Nuclear Energy, 27(16): 1483-1504, (2000).
  • [9] Atalay, M. A., “Milne problem for linearly anisotropic scattering and a specular and diffuse reflecting boundary”, Jornal of Quantative Spectroscopy and Radiative Transfer, 72(5): 589-606, (2002).
  • [10] Hussein, A., and Selim, M.M., “A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion”, Journal of Quantitative Spectroscopy and Radiative Transfer, 125: 84-92, (2013).
  • [11] Slama, H., El-Bedwhey, N.A., El-Depsy, A. and Selim, M. M., “Solution of the finite Milne problem in stochastic media with RVT Technique”, The European Physical Journal Plus, 132: 505, (2017).
  • [12] Hussein. A., and Selim, M. M., “A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique”, J Jornal of Quantative Spectroscopy and Radiative Transfer, 232: 54-65, (2019).
  • [13] Sahni, D. C., Türeci, R. G., and Bozkir, A. Z. “Partial Range Completeness of Case Eigenfunctions and Numerical Solution of Singular Integral Equations of Particle Transport Problems”, Journal of Computational and Theoretical Transport. 49(7): 349–367, (2020).
  • [14] Ganapol, B. D., “The Ln/Ln0 Method for 1D Neutron Transport in a Slab Medium”, Journal of Computational and Theoretical Transport, 51(5): 239-264, (2022).
  • [15] Ganapol , B. D., “Numerical Caseology by Lagrange Interpolation for the 1D Neutron Transport Equation in a Slab”, Nuclear Science and Engineering, 197(1): 1-13, (2023).
  • [16] Siewert, C. E., “The FN method for solving radiative-transfer problems in plane geometry”, Astrophysical Space Science. 58 (1): 131–7, (1978).
  • [17] Tezcan, C., Kaşkaş, A., and Güleçyüz, M. Ç., 2003. “The HN method for solving linear transport equation: Theory and applications”, Jornal of Quantative Spectroscopy and Radiative Transfer, 78 (2): 243-254, (2003).
  • [18] Gülderen, D., Sahni, D. C., Türeci, R. G., and Aydιn, A.,“The Milne Problem for Linear-Triplet Anisotropic Scattering with HN Method”, Journal of Computational and Theoretical Transport, 51(6): 329-353, (2022).
  • [19] Türeci, R. G., “The Milne Problem with the Anlı-Güngör Scattering”, Journal of Computational and Theoretical Transport, 51(6): 354-371, (2022).
  • [20] Numpy, https://numpy.org/. Access date: 22.07.2024
  • [21] Pandas, https://pandas.pydata.org/. Access date: 22.07.2024
  • [22] Seaborn, https://seaborn.pydata.org/. Access date: 22.07.2024
  • [23] Matplotlib, https://matplotlib.org/. Access date: 22.07.2024
  • [24] Yeo, I. K., and Richard A. J., “A New Family of Power Transformations to Improve Normality or Symmetry”, Biometrika, 87 (4): 954–959, (2000).
  • [25] Sklearn, https://scikit-learn.org/stable/. Access date: 22.07.2024
  • [26] Nair, V., and Hinton, G. E., “Rectified Linear Units Improve Restricted Boltzmann Machines”, Conference: Proceedings of the 27th International Conference on Machine Learning (ICML-10), June 21-24, Haifa, Israel, (2010).
  • [27] Keras, https://keras.io/. Access date: 22.07.2024
  • [28] Diederik P. K., Jimmy B.,“Adam: A Method for Stochastic Optimization”, Computer Science > Machine Learning, arXiv:1412.6980, (2014).
  • [29] Zheng C., Liub L., and Muc L., “Solving the linear transport equation by a deep neural network approach”, Journal of Discrete and Continuous Dynamical System-S, 15(4): 669-686, (2021).
  • [30] Xie Y., Wang Y., Ma Y., and Wu Z., “Neural Network Based Deep Learning Method for Multi-Dimensional Neutron Diffusion Problems with Novel Treatment to Boundary”, Journal of Nuclear Engineering, 2: 533-552, (2021).
  • [31] Whewell, B., and McClarren, R. G., “Data reduction in deterministic neutron transport calculations using machine learning”, Annals of Nuclear Energy, 176, 109276, (2022).
  • [32] Zolfaghari M., Masoudi S. F., Rahmani F., Fathi A., “Thermal neutron beam optimization for PGNAA applications using Q-learning algorithm and neural network”, Scientific Reports, 12, 8635, (2022). [33] Türeci R. G., “Machine Learning Applications to the One-speed Neutron Transport Problems”, Cumhuriyet Science Journal, 43(4): 726-738. (2022).

Yıl 2025, Erken Görünüm, 1 - 1

R. Gökhan Türeci

https://doi.org/10.35378/gujs.1521834

Öz

Kaynakça

  • [1] Placzek, G., and W. Seidel., “Milne's Problem in Transport Theory”, Physical Review, 72: 550-555, (1947).
  • [2] Noble, B., The Wiener-Hopf Technique, Pergamon Press, Oxford, (1958).
  • [3] Case, K.M. and Zweifel, P.F., Linear Transport Theory, Massachusetts: Addition-Wesley, (1967).
  • [4] McCormick, N.J., “One-Speed Neutron Transport Problems in Plane Geometry”, PhD thesis, The University of Michigan, (1964).
  • [5] McCormick N. J., and Kuščer, I., “Half‐Space Neutron Transport with Linearly Anisotropic Scattering”, Journal of Mathematical Physics, 6 (12): 1939-1945, (1964).
  • [6] Kavenoky, A., “The CN Method of Solving the Transport Equation: Application to Plane Geometry”, Nuclear Science and Engineering, 65, 209, (1978).
  • [7] Sahni, D.C., and Kumar, V., “Numerical solution of singular integral equations of neutron transport problems”, Transport Theory and Statistical Physics, 16(7): 959-978, (1987).
  • [8] Atalay, M. A., “Milne problem for linearly anisotropic scattering and a specularly reflecting boundary”, Annals of Nuclear Energy, 27(16): 1483-1504, (2000).
  • [9] Atalay, M. A., “Milne problem for linearly anisotropic scattering and a specular and diffuse reflecting boundary”, Jornal of Quantative Spectroscopy and Radiative Transfer, 72(5): 589-606, (2002).
  • [10] Hussein, A., and Selim, M.M., “A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion”, Journal of Quantitative Spectroscopy and Radiative Transfer, 125: 84-92, (2013).
  • [11] Slama, H., El-Bedwhey, N.A., El-Depsy, A. and Selim, M. M., “Solution of the finite Milne problem in stochastic media with RVT Technique”, The European Physical Journal Plus, 132: 505, (2017).
  • [12] Hussein. A., and Selim, M. M., “A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique”, J Jornal of Quantative Spectroscopy and Radiative Transfer, 232: 54-65, (2019).
  • [13] Sahni, D. C., Türeci, R. G., and Bozkir, A. Z. “Partial Range Completeness of Case Eigenfunctions and Numerical Solution of Singular Integral Equations of Particle Transport Problems”, Journal of Computational and Theoretical Transport. 49(7): 349–367, (2020).
  • [14] Ganapol, B. D., “The Ln/Ln0 Method for 1D Neutron Transport in a Slab Medium”, Journal of Computational and Theoretical Transport, 51(5): 239-264, (2022).
  • [15] Ganapol , B. D., “Numerical Caseology by Lagrange Interpolation for the 1D Neutron Transport Equation in a Slab”, Nuclear Science and Engineering, 197(1): 1-13, (2023).
  • [16] Siewert, C. E., “The FN method for solving radiative-transfer problems in plane geometry”, Astrophysical Space Science. 58 (1): 131–7, (1978).
  • [17] Tezcan, C., Kaşkaş, A., and Güleçyüz, M. Ç., 2003. “The HN method for solving linear transport equation: Theory and applications”, Jornal of Quantative Spectroscopy and Radiative Transfer, 78 (2): 243-254, (2003).
  • [18] Gülderen, D., Sahni, D. C., Türeci, R. G., and Aydιn, A.,“The Milne Problem for Linear-Triplet Anisotropic Scattering with HN Method”, Journal of Computational and Theoretical Transport, 51(6): 329-353, (2022).
  • [19] Türeci, R. G., “The Milne Problem with the Anlı-Güngör Scattering”, Journal of Computational and Theoretical Transport, 51(6): 354-371, (2022).
  • [20] Numpy, https://numpy.org/. Access date: 22.07.2024
  • [21] Pandas, https://pandas.pydata.org/. Access date: 22.07.2024
  • [22] Seaborn, https://seaborn.pydata.org/. Access date: 22.07.2024
  • [23] Matplotlib, https://matplotlib.org/. Access date: 22.07.2024
  • [24] Yeo, I. K., and Richard A. J., “A New Family of Power Transformations to Improve Normality or Symmetry”, Biometrika, 87 (4): 954–959, (2000).
  • [25] Sklearn, https://scikit-learn.org/stable/. Access date: 22.07.2024
  • [26] Nair, V., and Hinton, G. E., “Rectified Linear Units Improve Restricted Boltzmann Machines”, Conference: Proceedings of the 27th International Conference on Machine Learning (ICML-10), June 21-24, Haifa, Israel, (2010).
  • [27] Keras, https://keras.io/. Access date: 22.07.2024
  • [28] Diederik P. K., Jimmy B.,“Adam: A Method for Stochastic Optimization”, Computer Science > Machine Learning, arXiv:1412.6980, (2014).
  • [29] Zheng C., Liub L., and Muc L., “Solving the linear transport equation by a deep neural network approach”, Journal of Discrete and Continuous Dynamical System-S, 15(4): 669-686, (2021).
  • [30] Xie Y., Wang Y., Ma Y., and Wu Z., “Neural Network Based Deep Learning Method for Multi-Dimensional Neutron Diffusion Problems with Novel Treatment to Boundary”, Journal of Nuclear Engineering, 2: 533-552, (2021).
  • [31] Whewell, B., and McClarren, R. G., “Data reduction in deterministic neutron transport calculations using machine learning”, Annals of Nuclear Energy, 176, 109276, (2022).
  • [32] Zolfaghari M., Masoudi S. F., Rahmani F., Fathi A., “Thermal neutron beam optimization for PGNAA applications using Q-learning algorithm and neural network”, Scientific Reports, 12, 8635, (2022). [33] Türeci R. G., “Machine Learning Applications to the One-speed Neutron Transport Problems”, Cumhuriyet Science Journal, 43(4): 726-738. (2022).
Toplam 32 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Yapay Zeka (Diğer), Nükleer ve Plazma Fiziği (Diğer)
Bölüm Research Article
Yazarlar

R. Gökhan Türeci 0000-0001-6309-6300

Erken Görünüm Tarihi 3 Ocak 2025
Yayımlanma Tarihi
Gönderilme Tarihi 24 Temmuz 2024
Kabul Tarihi 3 Aralık 2024
Yayımlandığı Sayı Yıl 2025 Erken Görünüm

Kaynak Göster

APA Türeci, R. G. (2025). Deep Learning Application for Milne problem with linear anisotropic scattering. Gazi University Journal of Science1-1. https://doi.org/10.35378/gujs.1521834
AMA Türeci RG. Deep Learning Application for Milne problem with linear anisotropic scattering. Gazi University Journal of Science. Published online 01 Ocak 2025:1-1. doi:10.35378/gujs.1521834
Chicago Türeci, R. Gökhan. “Deep Learning Application for Milne Problem With Linear Anisotropic Scattering”. Gazi University Journal of Science, Ocak (Ocak 2025), 1-1. https://doi.org/10.35378/gujs.1521834.
EndNote Türeci RG (01 Ocak 2025) Deep Learning Application for Milne problem with linear anisotropic scattering. Gazi University Journal of Science 1–1.
IEEE R. G. Türeci, “Deep Learning Application for Milne problem with linear anisotropic scattering”, Gazi University Journal of Science, ss. 1–1, Ocak 2025, doi: 10.35378/gujs.1521834.
ISNAD Türeci, R. Gökhan. “Deep Learning Application for Milne Problem With Linear Anisotropic Scattering”. Gazi University Journal of Science. Ocak 2025. 1-1. https://doi.org/10.35378/gujs.1521834.
JAMA Türeci RG. Deep Learning Application for Milne problem with linear anisotropic scattering. Gazi University Journal of Science. 2025;:1–1.
MLA Türeci, R. Gökhan. “Deep Learning Application for Milne Problem With Linear Anisotropic Scattering”. Gazi University Journal of Science, 2025, ss. 1-1, doi:10.35378/gujs.1521834.
Vancouver Türeci RG. Deep Learning Application for Milne problem with linear anisotropic scattering. Gazi University Journal of Science. 2025:1-.