Abstract
A method for constructing multilayer neural network approximations of solutions of differential equations, based on the finite difference method, is proposed. The advantage of the method is the possibility of obtaining a neural network model of arbitrarily high accuracy without a time-consuming learning procedure. The solution is given by an analytical expression, explicitly including the parameters of the problem. The resulting neural network can, if necessary, be retrained according to the usual algorithm. The method is illustrated by the example of solving a particular ordinary second-order differential equation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice Hall, Upper Saddle River (1999)
Lagaris, I.E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9(5), 987–1000 (1998)
Dissanayake, M.W.M.G., Phan-Thien, N.: Neural-network-based approximations for solving partial differential equations. Commun. Numer. Methods Eng. 10(3), 195–201 (1994)
Fasshauer, G.E.: Solving differential equations with radial basis functions: multilevel methods and smoothing. Adv. Comput. Math. 11, 139–159 (1999)
Fornberg, B., Larsson, E.A.: Numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46, 891–902 (2003)
Galperin, E., Pan, Z., Zheng, Q.: Application of global optimization to implicit solution of partial differential equations. Comput. Math. Appl. 25(10/11), 119–124 (1993)
Galperin, E., Zheng, Q.: Solution and control of PDE via global optimization methods. Comput. Math. Appl. 25(10/11), 103–118 (1993)
Sharan, M., Kansa, E.J., Gupta, S.: Application of the multiquadric method to the numerical solution of elliptic partial differential equations. Appl. Math. Comput. 84, 275–302 (1997)
Vasilyev, A., Tarkhov, D., Guschin, G.: Neural networks method in pressure gauge modeling. In: Proceedings of the 10th IMEKO TC7 International Symposium on Advances of Measurement Science, Saint-Petersburg, Russia, vol. 2, pp. 275–279 (2004)
Tarkhov, D., Vasilyev, A.: New neural network technique to the numerical solution of mathematical physics problems. I: simple problems. Opt. Mem. Neural Netw. (Inf. Opt.) 14(1), 59–72 (2005)
Tarkhov, D., Vasilyev, A.: New neural network technique to the numerical solution of mathematical physics problems. II: complicated and nonstandard problems. Opt. Mem. Neural Netw. (Inf. Opt.) 14(2), 97–122 (2005)
Vasilyev, A., Tarkhov, D.: Mathematical models of complex systems on the basis of artificial neural networks. Nonlinear Phenom. Complex Syst. 17(2), 327–335 (2014)
Shemyakina, T.A., Tarkhov, D.A., Vasilyev, A.N.: Neural network technique for processes modeling in porous catalyst and chemical reactor. In: Cheng, L., Liu, Q., Ronzhin, A. (eds.) ISNN 2016. LNCS, vol. 9719, pp. 547–554. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40663-3_63
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. In: Cheng, L., Liu, Q., Ronzhin, A. (eds.) ISNN 2016. LNCS, vol. 9719, pp. 310–316. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40663-3_36
Budkina, E.M., Kuznetsov, E.B., Lazovskaya, T.V., Leonov, S.S., Tarkhov, D.A., Vasilyev, A.N.: Neural network technique in boundary value problems for ordinary differential equations. In: Cheng, L., Liu, Q., Ronzhin, A. (eds.) ISNN 2016. LNCS, vol. 9719, pp. 277–283. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40663-3_32
Lozhkina, O., Lozhkin, V., Nevmerzhitsky, N., Tarkhov, D., Vasilyev, A.: Motor transport related harmful PM2.5 and PM10: from on road measurements to the modelling of air pollution by neural network approach on street and urban level. J. Phys.: Conf. Ser. 772 (2016). http://iopscience.iop.org/article/10.1088/1742–6596/772/1/012031
Kaverzneva, T., Lazovskaya, T., Tarkhov, D., Vasilyev, A.: Neural network modeling of air pollution in tunnels according to indirect measurements. J. Phys.: Conf. Ser. 772 (2016). http://iopscience.iop.org/article/10.1088/1742-6596/772/1/012035
Lazovskaya, T.V., Tarkhov, D.A., Vasilyev, A.N.: Parametric neural network modeling in engineering. Recent Patents Eng. 11(1), 10–15 (2017)
Antonov, V., Tarkhov, D., Vasilyev, A.: Unified approach to constructing the neural network models of real objects. Part 1. Math. Models Methods Appl. Sci. 41(18), 9244–9251 (2018)
Lazovskaya, T., Tarkhov, D.: Multilayer neural network models based on grid methods. In: IOP Conference Series: Materials Science and Engineering, vol. 158 (2016). http://iopscience.iop.org/article/10.1088/1757-899X/158/1/01206
Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problem. Springer, Berlin (1987). https://doi.org/10.1007/978-3-662-12607-3
Bolgov, I., Kaverzneva, T., Kolesova, S., Lazovskaya, T., Stolyarov, O., Tarkhov, D.: Neural network model of rupture conditions for elastic material sample based on measurements at static loading under different strain rates. J. Phys.: Conf. Ser. 772 (2016). http://iopscience.iop.org/article/10.1088/1742-6596/772/1/012032
Aranda-Iglesias, D., Vadillo, G., Rodríguez-Martínez, J.A., Volokh, K.Y.: Modeling deformation and failure of elastomers at high strain rates. Mech. Mater. 104, 85–92 (2017)
Zéhil, G.-P., Gavin, H.P.: Unified constitutive modeling of rubber-like materials under diverse loading conditions. Int. J. Eng. Sci. 62, 90–105 (2013)
Hearle, J.W.S.: One-Dimensional Textiles. Handbook of Technical Textiles. Elsevier, Amsterdam (2016)
McKenna, H.A., Hearle, J.W.S., O’Hear, N.: Handbook of Fibre Rope Technology. Handbook of Fibre Rope Technology. Elsevier, Amsterdam (2004)
Weller, S.D., Johanning, L., Davies, P., Banfield, S.J.: Synthetic mooring ropes for marine renewable energy applications. Renew. Energy 83, 1268–1278 (2015)
Vasilyev, A.N., Tarkhov, D.A., Tereshin, V.A., Berminova, M.S., Galyautdinova, A.R.: Semi-empirical neural network model of real thread sagging. In: Kryzhanovsky, B., Dunin-Barkowski, W., Redko, V. (eds.) NEUROINFORMATICS 2017. SCI, vol. 736, pp. 138–144. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-66604-4_21
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. J. Phys.: Conf. Ser. 1044 (2018). https://iopscience.iop.org/article/10.1088/1742-6596/1044/1/012005/pdf
Bortkovskaya, M.R., et al.: Modeling of the membrane bending with multilayer semi-empirical models based on experimental data. In: Proceedings of the 2nd International Scientific Conference “Convergent Cognitive Information Technologies” (Convergent’2017) (2017). http://ceur-ws.org/Vol-2064/paper18.pdf
Acknowledgments
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Kaverzneva, T.T., Malykhina, G.F., Tarkhov, D.A. (2019). From Differential Equations to Multilayer Neural Network Models. In: Lu, H., Tang, H., Wang, Z. (eds) Advances in Neural Networks – ISNN 2019. ISNN 2019. Lecture Notes in Computer Science(), vol 11554. Springer, Cham. https://doi.org/10.1007/978-3-030-22796-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-22796-8_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22795-1
Online ISBN: 978-3-030-22796-8
eBook Packages: Computer ScienceComputer Science (R0)