Sampling of integrand for integration using shallow neural network

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Abstract

Inthispaper,westudytheeffectofusingtheMetropolis-Hastingsalgorithmforsamplingtheintegrand on the accuracy of calculating the value of the integral with the use of shallow neural network. In addition, a hybrid method for sampling the integrand is proposed, in which part of the training sample is generated by applying the Metropolis-Hastings algorithm, and the other part includes points of a uniform grid. Numerical experiments show that when integrating in high-dimensional domains, sampling of integrands both by the Metropolis-Hastings algorithm and by a hybrid method is more efficient with respect to the use of a uniform grid.

About the authors

Alexander S. Ayriyan

Joint Institute for Nuclear Research; Alikhanyan National Science Laboratory; Dubna State University

Email: ayriyan@jinr.ru
ORCID iD: 0000-0002-5464-4392

PhD in Physics and Mathematics, Head of sector of the Division of Computational Physics of JINR, Assistant professor of Department of Distributed Information Computing Systems of Dubna State University; Senior Researcher of AANL

6 Joliot-Curie St, Dubna, 141980, Russian Federation; 2 Alikhanyan Brothers St, Yerevan, 0036, Republic of Armenia; 19 Universitetskaya St, Dubna, 141980, Russian Federation

Hovik A. Grigorian

Joint Institute for Nuclear Research; Alikhanyan National Science Laboratory; Dubna State University; Yerevan State University

Email: hovik.grigorian@gmail.com
ORCID iD: 0000-0002-0003-0512

Candidate of Physical and Mathematical Sciences, Senior Researcher of JINR; Senior Researcher of AANL (YerPhI); Assistant professor of Dubna State University; assistant professor of Yerevan State University

6 Joliot-Curie St, Dubna, 141980, Russian Federation; 2 Alikhanyan Brothers St, Yerevan, 0036, Republic of Armenia; 19 Universitetskaya St, Dubna, 141980, Russian Federation; 1 Alex Manoogian St, Yerevan, 0025, Republic of Armenia

Vladimir V. Papoyan

Joint Institute for Nuclear Research; Alikhanyan National Science Laboratory; Dubna State University

Author for correspondence.
Email: vlpapoyan@jinr.ru
ORCID iD: 0000-0003-0025-5444

Junior researcher of JINR, Junior researcher of AANL (YerPhI), PhD student of Dubna State University

6 Joliot-Curie St, Dubna, 141980, Russian Federation; 2 Alikhanyan Brothers St, Yerevan, 0036, Republic of Armenia; 19 Universitetskaya St, Dubna, 141980, Russian Federation

References

  1. Lloyd, S., Irani, R. A. & Ahmadi, M. Using neural networks for fast numerical integration and optimization. IEEE Access 8, 84519-84531. doi: 10.1109/access.2020.2991966 (2020).
  2. Cybenko, G. Approximation by superpositions of a sigmoidal function. Mathematics of Control Signals and Systems 2, 303-314. doi: 10.1007/bf02551274 (Dec. 1989).
  3. Hastings, W. K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97-109. doi: 10.1093/biomet/57.1.97 (Apr. 1970).
  4. Chib, S. & Greenberg, E. Understanding the Metropolis-Hastings algorithm. The American Statistician 49, 327. doi: 10.2307/2684568 (Nov. 1995).
  5. Ecosystem for tasks of machine learning, deep learning and data analysis http://hlit.jinr.ru/en/access-to-resources_eng/ecosystem-for-ml_dl_bigdataanalysistasks_eng/. Accessed: 2023-10-10.
  6. Chollet, F. et al. Keras https://keras.io.
  7. Johansson, F. et al. mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.18) http://mpmath.org.
  8. Marquardt, D. W. An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics 11, 431-441. doi: 10.1137/0111030 (June 1963).
  9. Marco, F. D. Tensorflow Levenberg-Marquardt https://github.com/fabiodimarco/tflevenberg-marquardt.
  10. Kişi, Ö. & Uncuoǧlu, E. Comparison of three back-propagation training algorithms for two case studies. Indian Journal of Engineering and Materials Sciences 12, 434-442 (Oct. 2005).
  11. Jiawei Han, M. K. & Pei, J. Data mining: concepts and techniques Third Edition. 703 pp. doi: 10.1016/c2009-0-61819-5 (Elsevier Inc., 225 Wyman Street, Waltham, MA 02451, USA, 2012).
  12. Genz, A. A package for testing multiple integration subroutines in Numerical Integration 337-340 (Springer Netherlands, 1987). doi: 10.1007/978-94-009-3889-2_33.

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