Abstract
We consider the problems arising in the construction of the solutions of singularly perturbed differential equations. Usually, the decision of such problems by standard methods encounters significant difficulties of various kinds. The use of a common neural network approach is demonstrated in three model problems for ordinary differential equations. The conducted computational experiments confirm the effectiveness of this approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Tarkhov, D. and Vasilyev, A., New neural network technique to the numerical solution of mathematical physics problems, I: Simple problems, Opt. Mem. Neural Networks (Inform. Optics), 2005, vol. 14, pp. 59–72.
Tarkhov, D. and Vasilyev, A., New neural network technique to the numerical solution of mathematical physics problems, II: Complicated and nonstandard problems, Opt. Mem. Neural Networks (Inform. Optics), 2005, vol. 14, pp. 97–122.
Vasilyev, A.N. and Tarkhov, D.A., Mathematical models of complex systems on the basis of artificial neural networks, Nonlinear Phenomena in Complex Systems, 2014, vol. 17, no. 3, pp. 327–335.
Kainov, N.U., Tarkhov, D.A., and Shemyakina, T.A., Application of neural network modeling to identication and prediction problems in ecology data analysis for metallurgy and welding industry, Nonlinear Phenomena in Complex Systems, 2014, vol. 17, no. 1, pp. 57–63.
Lazovskaya, T.V. and Tarkhov, D.A., Fresh approaches to the construction of parameterized neural network solutions of a stiff differential equation, St. Petersburg Polytech. Univ. J.: Phys. Math., 2015. http://dx.doi.org/. doi 10.1016/j.spjpm.2015.07.005
Budkina, E.M., Kuznetsov, E.B., Lazovskaya, T.V., Leonov, S.S., Tarkhov, D.A., and Vasilyev, A.N., Neural Network Technique in Boundary Value Problems for Ordinary Differential Equations, Switzerland: Springer International Publishing, 2016; Cheng, L. et al., Eds., ISNN 2016, LNCS 9719, 2016, pp. 277–283.
Gorbachenko, V.I., Lazovskaya, T.V., Tarkhov, D.A., Vasilyev, A.N., and Zhukov, M.V., Neural Network Technique in Some Inverse Problems of Mathematical Physics, Switzerland: Springer International Publishing, 2016; Cheng L. et al., Eds., ISNN 2016, LNCS 9719, 2016, pp. 320–316.
Shemyakina, T.A., Tarkhov, D.A., and Vasilyev, A.N., Neural Network Technique for Processes Modeling in Porous Catalyst and Chemical Reactor, Switzerland: Springer International Publishing, 2016; Cheng L. et al., Eds., ISNN 2016, LNCS 9719, 2016, pp. 547–554.
Khudyaev, S.I., Porogovye yavleniya v nelineinyh uravneniyah, Moscow, Nauka, 2003, p. 268.
Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer, 2010, p. 614.
Riedmiller, M. and Braun, H., A direct adaptive method for faster backpropagation learning: The Rprop algorithm, Proceedings of the IEEE International Conference on Neural Networks, IEEE Press, 1993, pp. 586–591.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Budkina, E.M., Kuznetsov, E.B., Lazovskaya, T.V. et al. Neural network approach to intricate problems solving for ordinary differential equations. Opt. Mem. Neural Networks 26, 96–109 (2017). https://doi.org/10.3103/S1060992X17020011
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1060992X17020011