МЕТОДЫ СОЗДАНИЯ ЦИФРОВЫХ ДВОЙНИКОВ НА ОСНОВЕ НЕЙРОСЕТЕВОГО МОДЕЛИРОВАНИЯ
Аннотация
Предполагается, что к 2021 году около половины компаний будут использовать цифровых двойников разного уровня. Самые простые цифровые модели-двойники могут не использовать машинное обучение, но наибольшее преимущество будут иметь модели, использующие алгоритмы машинного обучения. В данной статье мы предлагаем свой подход к построению цифровых двойников реальных объектов. Мы опираемся на свой унифицированный процесс построения приближённых решений краевых задач для уравнений математической физики и накопленный опыт решения большого числа конкретных задач такого типа. В работе мы представляем пять подходов к построению цифровых моделей-двойников, опирающиеся на разработанные и протестированные нами эволюционные алгоритмы. Особенностью нашего подхода к эволюционным алгоритмам является использование генетических процедур для построения структуры модели и алгоритмов нелинейной оптимизации для подстройки ее параметров. Кроме того, мы предлагаем наш подход к построению многослойных моделей по дифференциальным уравнениям, позволяющий обойтись без трудоёмкой процедуры обучения нейронных сетей. Мы уверены, что предложенные нами подходы позволяют существенно упростить и унифицировать создание и адаптацию (поддержание в актуальном состоянии) цифровых двойников реальных объектов разного рода – технических, биологических, социально-экономических и т.д.
Литература
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[13] Marussig B., Zechner J., Beer G., Fries T.P. Fast isogeometric boundary element method based on independent field approximation. Computer Methods in Applied Mechanics and Engineering. 2015; 284:458-488. DOI: 10.1016/j.cma.2014.09.035
[14] Atroshchenko E., Xu G., Tomar S., Bordas S. Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub-and super-geometric analysis to Geometry Independent Field approximaTion (GIFT). 2017. arXiv:1706.06371
[15] Saputra A., Talebi H., Tran D., Birk C., Song C. Automatic image-based stress analysis by the scaled boundary finite element method. International Journal for Numerical Methods in Engineering. 2017; 109(5):697-738. DOI: 10.1002/nme.5304
[16] Ooi E., Song C., Natarajan S. A scaled boundary finite element formulation with bubble functions for elasto-static analyses of functionally graded materials. Computational Mechanics. 2017; 60(6):943–967. DOI: 10.1007/s00466-017-1443-y
[17] Natarajan S., Ooi E.T., Saputra A., Song C. A scaled boundary finite element formulation over arbitrary faceted star convex polyhedral. Engineering Analysis with Boundary Elements. 2017; 80:218-229. DOI: 10.1016/j.enganabound.2017.03.007
[18] Simpson R., Trevelyan J. A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics. Computer Methods in Applied Mechanics and Engineering. 2011; 200(1-4):1-10. DOI: 10.1016/j.cma.2010.06.015
[19] Peng X., Atroshchenko E., Kerfriden P., Bordas S. Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth. Computer Methods in Applied Mechanics and Engineering. 2017; 316:151-185. DOI: 10.1016/j.cma.2016.05.038
[20] Scott M.A., Simpson R.N., Evans J.A., Lipton S., Bordas S.P., Hughes T.J., Sederberg T.W. Isogeometric boundary element analysis using unstructured T-splines. Computer Methods in Applied Mechanics and Engineering. 2013; 254:197-221. DOI: 10.1016/j.cma.2012.11.001
[21] Oden J.T., Lima E.A., Almeida R.C., Feng Y., Rylander M.N., Fuentes D., Faghihi D., Rahman M.M., DeWitt M., Gadde M., Cliff Z.J. Toward predictive multiscale modeling of vascular tumor growth. Archives of Computational Methods in Engineering. 2016; 23(4):735-779. DOI: 10.1007/s11831-015-9156-x
[22] Fuentes D., Oden J., Diller K., Hazle J., Elliott A., Shetty A., Stafford R. Computational modeling and real-time control of patient-specific laser treatment of cancer. Annals of Biomedical Engineering. 2009; 37(4):763-782. DOI: 10.1007/s10439-008-9631-8
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[24] Le Folgoc L., Delingette H., Criminisi A., Ayache N. Sparse Bayesian registration. P. Golland, N. Hata, C. Barillot, J. Hornegger, R. Howe (Eds.) MICCAI - Medical Image Computing and Computer Assisted Intervention. Proceedings of the 7th International Conference, Boston, MA, USA, September 14-18, 2014. Part I. Lecture Notes in Computer Science - LNCS. Vol. 8673. Springer, Cham, 2014. Pp. 235-242. DOI: 10.1007/978-3-319-10404-1\_30
[25] Haykin S. Neural Networks: A Comprehensive Foundation. Prentice Hall, 1999. 823 p.
[26] Vasilyev A., Tarkhov D. Mathematical Models of Complex Systems on the Basis of Artificial Neural Networks. Nonlinear Phenomena in Complex Systems, 2014; 17(3):327-335.
[27] Tarkhov D.A., Vasilyev A.N. New neural network technique to the numerical solution of mathematical physics problems. I: Simple problems. Optical Memory and Neural Networks (Information Optics). 2005; 14(1):59-72.
[28] Tarkhov D.A., Vasilyev A.N. New neural network technique to the numerical solution of mathematical physics problems. II: Complicated and nonstandard problems. Optical Memory and Neural Networks (Information Optics). 2005; 14(2):97-122.
[29] Gorbachenko V.I., Lazovskaya T.V., Tarkhov D.A., Vasilyev A.N., Zhukov M.V. Neural Network Technique in Some Inverse Problems of Mathematical Physics. L. Cheng, Q. Liu, A. Ronzhin (Eds.) Advances in Neural Networks – ISNN 2016. ISNN 2016. Lecture Notes in Computer Science - LNCS. Vol. 9719. Springer, Cham, 2016. Pp. 310-316. DOI: 10.1007/978-3-319-40663-3_36
[30] Shemyakina T.A., Tarkhov D.A., Vasilyev A.N. Neural Network Technique for Processes Modeling in Porous Catalyst and Chemical Reactor. L. Cheng, Q. Liu, A. Ronzhin (Eds.). Advances in Neural Networks – ISNN 2016. ISNN 2016. Lecture Notes in Computer Science - LNCS. Vol. 9719. Springer, Cham, 2016. Pp. 547-554. DOI: 10.1007/978-3-319-40663-3_63
[31] Kainov N.U., Tarkhov D.A., Shemyakina T.A. Application of Neural Network Modeling to Identification and Prediction Problems in Ecology Data Analysis for Metallurgy and Welding Industry. Nonlinear Phenomena in Complex Systems. 2014; 17(1):57–63.
[32] Kaverzneva T., Lazovskaya T., Tarkhov D., Vasilyev A. Neural network modeling of air pollution in tunnels according to indirect measurements. Journal of Physics: Conference Series. 2016; 772(1):012035. DOI: 10.1088/1742-6596/772/1/012035
[33] Lazovskaya T.N., Tarkhov D.A., Vasilyev A.N. Parametric Neural Network Modeling in Engineering. Recent Patents on Engineering. 2017; 11(1):10-15. DOI: 10.2174/1872212111666161207155157
[34] Budkina E.M., Kuznetsov E.B., Lazovskaya T.V., Tarkhov D.A., Shemyakina T.A., Vasilyev A.N. Neural network approach to intricate problems solving for ordinary differential equation. Optical Memory and Neural Networks. 2017; 26(2):96-109. DOI: 10.3103/S1060992X17020011
[35] Antonov V., Tarkhov D., Vasilyev A. Unified approach to constructing the neural network models of real objects. Part 1. Mathematical Methods in the Applied Sciences. 2018; 41(18):9244-9251. DOI: 10.1002/mma.5205
[36] Ivakhnenko A.G., Yurachkovsky Yu.P. Simulation of complex systems from experimental data [Modelirovanie slozhnyh sistem po ehksperimental'nym dannym]. M.: Radio and Communication, 1987. 120 p. (In Russian).
[37] Rastrigin L.A., Erenshtein R.H. Method of collective recognition [Metod kollektivnogo raspoznavanija]. M.: Energoizdat, 1981. 80 p. (In Russian)
[38] Lazovskaya T., Tarkhov D. Multilayer neural network models, based on grid methods. IOP Conference Series: Materials Science and Engineering. 2016; 158(1):012061. DOI: 10.1088/1757-899X/158/1/012061
[39] Lazovskaya T., Tarkhov D., Vasilyev А. Multi-Layer Solution of Heat Equation. B. Kryzhanovsky, W. Dunin-Barkowski, V. Redko (Eds.) Advances in Neural Computation, Machine Learning, and Cognitive Research. Studies in Computational Intelligence. Vol. 736. Springer International Publishing, 2018. Pp. 17–22. DOI: 10.1007/978-3-319-66604-4_3
[40] Vasilyev A.N., Tarkhov D.A., Tereshin V.A., Berminova M.S., Galyautdinova A.R. Semi-empirical Neural Network Model of Real Thread Sagging. B. Kryzhanovsky, W. Dunin-Barkowski, V. Redko (Eds.) Advances in Neural Computation, Machine Learning, and Cognitive Research. Studies in Computational Intelligence. Vol. 736. Springer International Publishing, 2018. Pp. 138–146. DOI: 10.1007/978-3-319-66604-4_21
[41] Zulkarnay I.U., Kaverzneva T.T., Tarkhov D.A., Tereshin V.A., Vinokhodov T.V., Kapitsin D.R. A Two-layer Semi-Empirical Model of Nonlinear Bending of the Cantilevered Beam. IOP Conference Series: Journal of Physics: Conference Series. 2018; 1044(conf. 1):012005. DOI: 10.1088/1742-6596/1044/1/012005
[42] Bortkovskaya M.R., Vasilyev P.I., Zulkarnay I.U., Semenova D.A., Tarkhov D.A., Udalov P.P., Shishkina I.A. Modeling of the membrane bending with multilayer semi-empirical models based on experimental data. V. Sukhomlin, E. Zubareva, M. Shneps-Shneppe (Eds.) Proceedings of the 2nd International scientific conference "Convergent cognitive information technologies" (Convergent’2017). Moscow, Russia: November 24–26, 2017. CEUR Workshop Proceedings. Vol. 2064. Pp. 150-156. Available at: http://ceur-ws.org/Vol-2064/paper18.pdf (accessed 05.06.2018).
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[44] Tarkhov D., Shershneva E. Approximate analytical solutions of Mathieu's equations based on classical numerical methods. V. Sukhomlin, E. Zubareva, M. Shneps-Shneppe (Eds.) Proceedings of the XI International Scientific-Practical Conference “Modern Information Technologies and IT-Education” (SITITO 2016). Moscow, Russia, November 25-26. 2016. CEUR Workshop Proceedings. Vol. 1761. Pр. 356-362. Available at: http://ceur-ws.org/Vol-1761/paper46.pdf (accessed 05.06.2018). (In Russian)
[2] Uhlemann T.H.-J., Schock C., Lehmann C., Freiberger S., Steinhilper R. The Digital Twin: Demonstrating the Potential of Real Time Data Acquisition in Production Systems. Procedia Manufacturing. 2017; 9:113-120. DOI: 10.1016/j.promfg.2017.04.043
[3] Boschert S., Rosen R. Digital Twin – The Simulation Aspect. P. Hehenberger, D. Bradley (Eds.) Mechatronic Futures. Springer International Publishing, 2016. Pp. 59-74. DOI: 10.1007/978-3-319-32156-1_5
[4] Dong-Ki K. et al. A Study of Resource Management for Fault-Tolerant and Energy Efficient Cloud Datacenter. Y. Zhang, L. Peng, C-H. Youn (Eds.) Cloud Computing. 6th International Conference, CloudComp 2015, Daejeon, South Korea, October 28-29, 2015, Revised Selected Papers. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering. Vol. 167. Springer International Publishing, 2016. Pp. 22-29. DOI: 10.1007/978-3-319-38904-2_3
[5] Chawla V., Sogani P. Cloud Computing – The Future. A. Mantri, S. Nandi, G. Kumar, S. Kumar (Eds.) High Performance Architecture and Grid Computing. HPAGC 2011. Communications in Computer and Information Science, Vol. 169. Springer, Berlin, Heidelberg, 2011. Pp. 113-118. DOI: 10.1007/978-3-642-22577-2_15
[6] Strang G., Fix G.J. An Analysis of the Finite Element Method. Vol. 212. Prentice-Hall, Englewood Cliffs, NJ, 1973. 306 p.
[7] Hughes T.J. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Courier Corporation, North Chelmsford, 2012. 704 p.
[8] Dhatt G., Lefrançois E., Touzot G. Finite Element Method. London, Wiley-ISTE, 2012. 624 p.
[9] Bathe K.J. Finite Element Method. Wiley Encyclopedia of Computer Science and Engineering. London, Wiley Online Library, 2008. DOI: 10.1002/9780470050118.ecse159
[10] Nguyen V.P., Rabczuk T., Bordas S., Duflot M. Meshless methods: a review and computer implementation aspects. Mathematics and Computers in Simulation. 2008; 79(3):763-813. DOI: 10.1016/j.matcom.2008.01.003
[11] Nguyen V.P., Anitescu C., Bordas S.P., Rabczuk T. Isogeometric analysis: an overview and computer implementation aspect. Mathematics and Computers in Simulation. 2015; 117:89-116. DOI: 10.1016/j.matcom.2015.05.008
[12] Hughes T.J., Cottrell J.A., Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering. 2005; 194(39-41):4135-4195. DOI: 10.1016/j.cma.2004.10.008
[13] Marussig B., Zechner J., Beer G., Fries T.P. Fast isogeometric boundary element method based on independent field approximation. Computer Methods in Applied Mechanics and Engineering. 2015; 284:458-488. DOI: 10.1016/j.cma.2014.09.035
[14] Atroshchenko E., Xu G., Tomar S., Bordas S. Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub-and super-geometric analysis to Geometry Independent Field approximaTion (GIFT). 2017. arXiv:1706.06371
[15] Saputra A., Talebi H., Tran D., Birk C., Song C. Automatic image-based stress analysis by the scaled boundary finite element method. International Journal for Numerical Methods in Engineering. 2017; 109(5):697-738. DOI: 10.1002/nme.5304
[16] Ooi E., Song C., Natarajan S. A scaled boundary finite element formulation with bubble functions for elasto-static analyses of functionally graded materials. Computational Mechanics. 2017; 60(6):943–967. DOI: 10.1007/s00466-017-1443-y
[17] Natarajan S., Ooi E.T., Saputra A., Song C. A scaled boundary finite element formulation over arbitrary faceted star convex polyhedral. Engineering Analysis with Boundary Elements. 2017; 80:218-229. DOI: 10.1016/j.enganabound.2017.03.007
[18] Simpson R., Trevelyan J. A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics. Computer Methods in Applied Mechanics and Engineering. 2011; 200(1-4):1-10. DOI: 10.1016/j.cma.2010.06.015
[19] Peng X., Atroshchenko E., Kerfriden P., Bordas S. Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth. Computer Methods in Applied Mechanics and Engineering. 2017; 316:151-185. DOI: 10.1016/j.cma.2016.05.038
[20] Scott M.A., Simpson R.N., Evans J.A., Lipton S., Bordas S.P., Hughes T.J., Sederberg T.W. Isogeometric boundary element analysis using unstructured T-splines. Computer Methods in Applied Mechanics and Engineering. 2013; 254:197-221. DOI: 10.1016/j.cma.2012.11.001
[21] Oden J.T., Lima E.A., Almeida R.C., Feng Y., Rylander M.N., Fuentes D., Faghihi D., Rahman M.M., DeWitt M., Gadde M., Cliff Z.J. Toward predictive multiscale modeling of vascular tumor growth. Archives of Computational Methods in Engineering. 2016; 23(4):735-779. DOI: 10.1007/s11831-015-9156-x
[22] Fuentes D., Oden J., Diller K., Hazle J., Elliott A., Shetty A., Stafford R. Computational modeling and real-time control of patient-specific laser treatment of cancer. Annals of Biomedical Engineering. 2009; 37(4):763-782. DOI: 10.1007/s10439-008-9631-8
[23] Lê M., Delingett, H., Kalpathy-Cramer J., Gerstner E.R., Batchelor T., Unkelbach J., Ayache N. Bayesian Personalization of Brain Tumor Growth Model. A.F. Frangi, J. Hornegger, N. Navab, W.M. Wells (Eds.) MICCAI - Medical Image Computing and Computer Assisted Intervention. 2015, Oct 2015, Munich, Germany. Lecture Notes in Computer Science - LNCS. Vol. 9350. Springer, New York, 2015. Pp. 424–432. DOI: 10.1007/978-3-319-24571-3_51
[24] Le Folgoc L., Delingette H., Criminisi A., Ayache N. Sparse Bayesian registration. P. Golland, N. Hata, C. Barillot, J. Hornegger, R. Howe (Eds.) MICCAI - Medical Image Computing and Computer Assisted Intervention. Proceedings of the 7th International Conference, Boston, MA, USA, September 14-18, 2014. Part I. Lecture Notes in Computer Science - LNCS. Vol. 8673. Springer, Cham, 2014. Pp. 235-242. DOI: 10.1007/978-3-319-10404-1\_30
[25] Haykin S. Neural Networks: A Comprehensive Foundation. Prentice Hall, 1999. 823 p.
[26] Vasilyev A., Tarkhov D. Mathematical Models of Complex Systems on the Basis of Artificial Neural Networks. Nonlinear Phenomena in Complex Systems, 2014; 17(3):327-335.
[27] Tarkhov D.A., Vasilyev A.N. New neural network technique to the numerical solution of mathematical physics problems. I: Simple problems. Optical Memory and Neural Networks (Information Optics). 2005; 14(1):59-72.
[28] Tarkhov D.A., Vasilyev A.N. New neural network technique to the numerical solution of mathematical physics problems. II: Complicated and nonstandard problems. Optical Memory and Neural Networks (Information Optics). 2005; 14(2):97-122.
[29] Gorbachenko V.I., Lazovskaya T.V., Tarkhov D.A., Vasilyev A.N., Zhukov M.V. Neural Network Technique in Some Inverse Problems of Mathematical Physics. L. Cheng, Q. Liu, A. Ronzhin (Eds.) Advances in Neural Networks – ISNN 2016. ISNN 2016. Lecture Notes in Computer Science - LNCS. Vol. 9719. Springer, Cham, 2016. Pp. 310-316. DOI: 10.1007/978-3-319-40663-3_36
[30] Shemyakina T.A., Tarkhov D.A., Vasilyev A.N. Neural Network Technique for Processes Modeling in Porous Catalyst and Chemical Reactor. L. Cheng, Q. Liu, A. Ronzhin (Eds.). Advances in Neural Networks – ISNN 2016. ISNN 2016. Lecture Notes in Computer Science - LNCS. Vol. 9719. Springer, Cham, 2016. Pp. 547-554. DOI: 10.1007/978-3-319-40663-3_63
[31] Kainov N.U., Tarkhov D.A., Shemyakina T.A. Application of Neural Network Modeling to Identification and Prediction Problems in Ecology Data Analysis for Metallurgy and Welding Industry. Nonlinear Phenomena in Complex Systems. 2014; 17(1):57–63.
[32] Kaverzneva T., Lazovskaya T., Tarkhov D., Vasilyev A. Neural network modeling of air pollution in tunnels according to indirect measurements. Journal of Physics: Conference Series. 2016; 772(1):012035. DOI: 10.1088/1742-6596/772/1/012035
[33] Lazovskaya T.N., Tarkhov D.A., Vasilyev A.N. Parametric Neural Network Modeling in Engineering. Recent Patents on Engineering. 2017; 11(1):10-15. DOI: 10.2174/1872212111666161207155157
[34] Budkina E.M., Kuznetsov E.B., Lazovskaya T.V., Tarkhov D.A., Shemyakina T.A., Vasilyev A.N. Neural network approach to intricate problems solving for ordinary differential equation. Optical Memory and Neural Networks. 2017; 26(2):96-109. DOI: 10.3103/S1060992X17020011
[35] Antonov V., Tarkhov D., Vasilyev A. Unified approach to constructing the neural network models of real objects. Part 1. Mathematical Methods in the Applied Sciences. 2018; 41(18):9244-9251. DOI: 10.1002/mma.5205
[36] Ivakhnenko A.G., Yurachkovsky Yu.P. Simulation of complex systems from experimental data [Modelirovanie slozhnyh sistem po ehksperimental'nym dannym]. M.: Radio and Communication, 1987. 120 p. (In Russian).
[37] Rastrigin L.A., Erenshtein R.H. Method of collective recognition [Metod kollektivnogo raspoznavanija]. M.: Energoizdat, 1981. 80 p. (In Russian)
[38] Lazovskaya T., Tarkhov D. Multilayer neural network models, based on grid methods. IOP Conference Series: Materials Science and Engineering. 2016; 158(1):012061. DOI: 10.1088/1757-899X/158/1/012061
[39] Lazovskaya T., Tarkhov D., Vasilyev А. Multi-Layer Solution of Heat Equation. B. Kryzhanovsky, W. Dunin-Barkowski, V. Redko (Eds.) Advances in Neural Computation, Machine Learning, and Cognitive Research. Studies in Computational Intelligence. Vol. 736. Springer International Publishing, 2018. Pp. 17–22. DOI: 10.1007/978-3-319-66604-4_3
[40] Vasilyev A.N., Tarkhov D.A., Tereshin V.A., Berminova M.S., Galyautdinova A.R. Semi-empirical Neural Network Model of Real Thread Sagging. B. Kryzhanovsky, W. Dunin-Barkowski, V. Redko (Eds.) Advances in Neural Computation, Machine Learning, and Cognitive Research. Studies in Computational Intelligence. Vol. 736. Springer International Publishing, 2018. Pp. 138–146. DOI: 10.1007/978-3-319-66604-4_21
[41] Zulkarnay I.U., Kaverzneva T.T., Tarkhov D.A., Tereshin V.A., Vinokhodov T.V., Kapitsin D.R. A Two-layer Semi-Empirical Model of Nonlinear Bending of the Cantilevered Beam. IOP Conference Series: Journal of Physics: Conference Series. 2018; 1044(conf. 1):012005. DOI: 10.1088/1742-6596/1044/1/012005
[42] Bortkovskaya M.R., Vasilyev P.I., Zulkarnay I.U., Semenova D.A., Tarkhov D.A., Udalov P.P., Shishkina I.A. Modeling of the membrane bending with multilayer semi-empirical models based on experimental data. V. Sukhomlin, E. Zubareva, M. Shneps-Shneppe (Eds.) Proceedings of the 2nd International scientific conference "Convergent cognitive information technologies" (Convergent’2017). Moscow, Russia: November 24–26, 2017. CEUR Workshop Proceedings. Vol. 2064. Pp. 150-156. Available at: http://ceur-ws.org/Vol-2064/paper18.pdf (accessed 05.06.2018).
[43] Vasilyev A., Tarkhov D., Shemyakina T. Approximate analytical solutions of ordinary differential equations. V. Sukhomlin, E. Zubareva, M. Shneps-Shneppe (Eds.) Proceedings of the XI International Scientific-Practical Conference “Modern Information Technologies and IT-Education” (SITITO 2016). Moscow, Russia, November 25-26. 2016. CEUR Workshop Proceedings. Vol. 1761. Pр. 393-400. Available at: http://ceur-ws.org/Vol-1761/paper50.pdf (accessed 05.06.2018). (In Russian)
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Опубликована
2018-09-30
Как цитировать
ВАСИЛЬЕВ, Александр Николаевич; ТАРХОВ, Дмитрий Альбертович; МАЛЫХИНА, Галина Федоровна.
МЕТОДЫ СОЗДАНИЯ ЦИФРОВЫХ ДВОЙНИКОВ НА ОСНОВЕ НЕЙРОСЕТЕВОГО МОДЕЛИРОВАНИЯ.
Современные информационные технологии и ИТ-образование, [S.l.], v. 14, n. 3, p. 521-532, sep. 2018.
ISSN 2411-1473. Доступно на: <http://sitito.cs.msu.ru/index.php/SITITO/article/view/414>. Дата доступа: 14 june 2025
doi: https://doi.org/10.25559/SITITO.14.201803.521-532.
Раздел
Теоретические вопросы информатики, прикладной математики, компьютерных наук
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