Abstract
The tasks of constructing mathematical models from heterogeneous data, including differential equations, boundary and initial conditions, observational data and other information about the modeled object, are of great practical importance, in particular, in the construction of digital counterparts of complex technical objects. Especially relevant is the search for methods for constructing the above mathematical models in a situation where the physical model and, consequently, the differential equation is known with insufficient precision for modeling purposes. We have developed new methods for constructing mathematical models of the type mentioned above and check them on a model problem with real measurements. In this paper, we consider the solution of the problem of modeling the deflection of a loaded circular membrane, in the center of which the weight of a given mass is located. The accuracy of the models expressing the dependence of the deflection of the membrane from the distance to the center is compared. We constructed the first model on the basis of an analytical solution of the equation of equilibrium conditions. The second model was obtained with the help of the original modification of the refined Euler method. When constructing the second model, it is necessary to select the same number of coefficients as in the construction of the first model. We built the third model in the form of an output of a neural network. The coefficients of the models were selected from the data obtained experimentally. The resulting approximate accuracy models outperform the model based on the exact solution. The neural network model turned out to be the most accurate, but it requires the selection of a larger number of coefficients.
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References
Kortelainen, J.: Semantic data model for multibody system modelling. Espoo VTT Publications 766 (2011)
Glotzer, S., Kim, S., Cummings, P., Deshmukh, A., Head-Gordon, M., Karniadakis, G., Petzold, L., Sagui, C., Shinozuka, M.: International assessment of research and development in simulation-based engineering and science. WTEC panel report, World Technology Evaluation Center, Inc. WTEC (2009)
Peherstorfer, B., Willcox, K.: Dynamic data-driven reduced-order models. Comput. Methods Appl. Mech. Eng. 291, 21–41 (2015)
Rosenblatt, F.: The perceptron: a probabilistic model for information storage and organization in the brain. Psychol. Rev. 65(6), 386 (1958)
Maaten, L., Hinton, G.: Visualizing data using t-SNE. J. Mach. Learn. Res. 9, 2579–2605 (2008)
Suzuki, K.: Artificial Neural Networks: Methodological Advances and Biomedical Applications. INTECH Open Access Publisher (2011)
Largris, I.E., Likas, A.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Networks 9(5), 987–1000 (1998)
Lazovskaya, T., Tarkhov, D.: Multilayer neural network models based on grid methods. IOP Conf. Ser.: Mater. Sci. Eng. 158 (2016). http://iopscience.iop.org/article/10.1088/1757-899X/158/1/01206
Vasilyev, A.N., Tarkhov, D.A., Tereshin, V.A., Berminova, M.S., Galyautdinova, A.R.: Semi-empirical neural network model of real thread sagging. Studies in Computational Intelligence, vol. 736, pp. 138–146. Springer, New York (2018)
Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problem. Springer, Berlin (1987)
Lazovskaya, T.V., Tarkhov, D.A., Vasilyev, A.N.: Parametric neural network modeling in engineering. Recent. Pat.S Eng. 11(1), 10–15 (2017)
Lozhkina, O., Lozhkin, V., Nevmerzhitsky, N., Tarkhov, D., Vasilyev, A.: Motor transport related harmful PM2.5 and PM10: from onroad measurements to the modelling of air pollution by neural network approach on street and urban level. J. Phys.: Conf. Ser. 772 (2016)
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The article was prepared on the basis of scientific research carried out with the financial support of the Russian Science Foundation grant (project No. 18-19-00474).
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Takhov, D.A. et al. (2019). Semiempirical Model of the Real Membrane Bending. In: Kryzhanovsky, B., Dunin-Barkowski, W., Redko, V., Tiumentsev, Y. (eds) Advances in Neural Computation, Machine Learning, and Cognitive Research II. NEUROINFORMATICS 2018. Studies in Computational Intelligence, vol 799. Springer, Cham. https://doi.org/10.1007/978-3-030-01328-8_26
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DOI: https://doi.org/10.1007/978-3-030-01328-8_26
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