Abstract
A new approach to the construction of multilayer neural network approximate solutions for evolutionary partial differential equations is considered. The approach is based on the application of the recurrence relations of the Euler, Runge-Kutta, etc. methods to variable length intervals. The resulting neural-like structure can be considered as a generalization of a feedforward multilayer network or a recurrent Hopfield network. This analogy makes it possible to apply known methods to the refinement of the obtained solution, for example, the backpropagation algorithm. Earlier, a similar approach has been successfully used by the authors in the case of ordinary differential equations. Computational experiments are performed on one test problem for the one-dimensional (in terms of spatial variables) heat equation. Explicit formulas are derived for the dependence of the resulting neural network output on the number of layers. It was found that the error tends to zero with an increasing number of layers, even without the use of the network learning.
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
Tikhonov, A.N., Samarskiy, A.A.: Uravneniya Matematicheskoy Fiziki. Izda-tel’stvo MGU, Moscow (1999). (in russian)
Lazovskaya, T., Tarkhov, D.: Multilayer neural network models based on grid methods. IOP Conf. Ser.: Mater. Sci. Eng. 158(1) (2016). doi:10.1088/1757-899X/158/1/012061
Samarskiy, A.A.: Vvedeniye v chislennyye metody. Lan’, SPb (2005). (in russian)
Hardle, W., Kerkyacharian, G., Picard, D., Tsybakov, A.: Wavelets, Approximation and Statistical Applications. Seminaire Paris-Berlin (1997)
Vasilyev, A.N., Tarkhov, D.A.: Mathematical models of complex systems on the basis of artificial neural networks. Nonlinear Phenom. Complex Syst. 17(3), 327–335 (2014)
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., et al. (eds.) ISNN 2016. LNCS, vol. 9719, pp. 277–283. Springer International Publishing, Switzerland (2016)
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., et al. (eds.) ISNN 2016. LNCS, vol. 9719, pp. 310–316. Springer International Publishing, Switzerland (2016)
Shemyakina, T.A., Tarkhov, D.A., Vasilyev, A.N.: Neural network technique for processes modeling in porous catalyst and chemical reactor. In: Cheng, L., et al. (eds.) ISNN 2016. LNCS, vol. 9719, pp. 547–554. Springer International Publishing, Switzerland (2016)
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). doi: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)
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). doi:10.1088/1742-6596/772/1/012032
Filkin, V., Kaverzneva, T., Lazovskaya, T., Lukinskiy, E., Petrov, A., Stolyarov, O., Tarkhov, D.: Neural network modeling of conditions of destruction of wood plank based on measurements. J. Phys.: Conf. Ser. 772 (2016). doi:10.1088/1742-6596/772/1/012041
Vasilyev, A., Tarkhov, D., Bolgov, I., Kaverzneva, T., Kolesova, S., Lazovskaya, T., Lukinskiy, E., Petrov, A., Filkin, V.: Multilayer neural network models based on experimental data for processes of sample deformation and destruction. In: Selected Papers of Convergent 2016, Moscow, 25–26 November 2016, pp. 6–14 (2016)
Tarkhov, D., Shershneva, E.: Approximate analytical solutions of Mathieus equations based on classical numerical methods. In: Selected Papers of SITITO 2016, Moscow, 25–26 November 2016, pp. 356–362 (2016)
Vasilyev, A., Tarkhov, D., Shemyakina, T.: Approximate analytical solutions of ordinary differential equations. In: Selected Papers of SITITO 2016, Moscow, 25–26 November 2016, pp. 393–400 (2016)
Neha, Y., Anupam, Y., Manoj, K.: An Introduction to Neural Network Methods for Differential Equations. Springer Briefs in Applied Sciences and Technology Computational Intelligence (2015)
Galushkin, A.I.: Neyronnyye seti: osnovy teorii. Goryachaya liniya - telekom, Moscow (2010). (in Russian)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Lazovskaya, T., Tarkhov, D., Vasilyev, A. (2018). Multi-Layer Solution of Heat Equation. In: Kryzhanovsky, B., Dunin-Barkowski, W., Redko, V. (eds) Advances in Neural Computation, Machine Learning, and Cognitive Research. NEUROINFORMATICS 2017. Studies in Computational Intelligence, vol 736. Springer, Cham. https://doi.org/10.1007/978-3-319-66604-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-66604-4_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66603-7
Online ISBN: 978-3-319-66604-4
eBook Packages: EngineeringEngineering (R0)