Abstract
In this paper, new methods for solving mathematical modelling problems, based on the usage of normalized radial basis functions, are introduced. Meshfree computational algorithms for solving classical and inverse problems of mathematical physics are developed. The distinctive feature of these algorithms is the usage of moving functional basis, which allows us to adapt to solution particularities and to maintain high accuracy at relatively low computational cost. Specifics of neural network algorithms application to non-stationary problems of mathematical physics were indicated. The paper studies the matters of application of developed algorithms to identification problems. Analysis of solution results for representative problems of source components (and boundary conditions) identification in heat transfer equations illustrates that the elaborated algorithms obtain regularization qualities and allow us to maintain high accuracy in problems with considerable measurement errors.
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References
Vasilyev, A.N., Tarkhov, D.A.: Neural Network Modelling Principles, Algorithms, Applications. SPbSPU Publishing House, Saint-Petersburg (2009). 528 (in Russian)
Vasilyev, A.N., Osipov, V.P., Tarkhov, D.A.: Unified modelling process for physicotechnical objects with distributed constants. St. Petersburg State Polytechnical Univ. J. 3, 39–52 (2010). (in Russian)
Bugmann, G.: Normalized radial basis function networks. Neurocomputing (Special Issue on Radial Basis Function Networks) 20, 97–110 (1998)
Haykin, S.: Neural Networks: A Comprehensive Foundation, 2nd edn. Williams Publishing House, Jerusalem (2006). 1104 (in Russian)
Kolbin, I.S., Reviznikov, D.L.: Solving mathematical physics problems using normalized radial basis functions neural-like networks. Neurocomputers: Developing, Applications. Radiotekhnika vol. 2, pp. 12–19 (2012). (in Russian)
Kolbin, I.S., Reviznikov, D.L.: Solving inverse mathematical physics problems using normalized radial basis functions neural-like networks. Neurocomputers: developing, applications. Radiotekhnika 9, 3–11 (2013). (in Russian)
Panteleyev, A.V., Letova, T.A.: Optimization Techniques Through Examples and Tasks. Vysshaya shkola, Moscow (2005). 544 (in Russian)
Hager, W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16, 170–192 (2005)
Chai, Z., Shi, B.: A novel lattice boltzmann model for the poisson equation. Appl. Math. Mech. 32, 2050–2058 (2008)
Kolbin, I.S.: Software package for the solution of mathematical modelling problems using neural networks techniques. Program Eng. 2, 25–30 (2013). (in Russian)
Xie, O., Zhao, Z.: Identifying an unknown source in the Poisson equation by a modified Tikhonov regularization method. Int. J. Math. Comput. Sci. 6, 86–90 (2012)
Acknowledgements
The work was supported by the Russian Foundation for Basic Research, project numbers 14-01-00660 and 14-01-00733.
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Vasilyev, A.N., Kolbin, I.S., Reviznikov, D.L. (2016). Meshfree Computational Algorithms Based on Normalized Radial Basis Functions. In: Cheng, L., Liu, Q., Ronzhin, A. (eds) Advances in Neural Networks – ISNN 2016. ISNN 2016. Lecture Notes in Computer Science(), vol 9719. Springer, Cham. https://doi.org/10.1007/978-3-319-40663-3_67
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DOI: https://doi.org/10.1007/978-3-319-40663-3_67
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