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Mathematical models of systems with interval representation of parameters on the basis of geterogeneous neural networks. Temperature field continuation – nonclassical problem statement

Keywords:

A. N. Vasilyev – Dr.Sc. (Eng.), Professor, Department Higher Mathematics, Applied Mathematics and Mechanics Institute, St. Petersburg State Polytechnical University. E-mail: a.n.vasilyev@gmail.com
D. A. Tarkhov – Dr.Sc. (Eng.), Professor, Department Higher Mathematics, Applied Mathematics and Mechanics Institute, St. Petersburg State Polytechnical University. E-mail: dtarkhov@gmail.com


Neural networking technique with models based on partial differential equations is applied to known incorrect problems which solution by routine approaches has difficulties. An approximate solution to the problem is found as output of neural network with some prescribed architecture. Network weights are determined in the process of stepwise network training grounded on some error functional minimization in general case. We consider the case when system parameters (BVP ones) are given in some variation intervals. This paper is the third part of trinomial composition. Given initial and boundary conditions for heat conduction equation allow us to solve the problem “forward”’, and such a problem is well known as a correct one. From the formal point of view these conditions allow us to search for the solutions in the case of temperature conduction coefficient various values, but it is an ill-posed problem now. Neural network approach gives us an opportunity to get a stable approximate solution – regularization of the problem for interval parameter. This problem in classical statement was considered in the second part of the composition – NC, 2012, № 11, pp. 56–59. Another example of similar problem is an incorrect problem of temperature field evaluation according to approximately known point measurement data. Some regularization of this problem was realized by A. Samarsky and P. Vabishevich as a control problem via initial data reconstruction on the set of point data (boundary conditions are given). By the offered approach based on the neural network methodology both the solution of correct direct problem and regularization of incorrect inverse problem are constructed uniformly. This nonclassical statement of the problem is considered in this third part of the composition. The similar problem in the case of interval temperature conduction factor was solved via neurocomputing on growing neural networks too. Results of neurocomputing and some corresponding figures are given. Advantages of neural network approach and some possible generalizations are mentioned.
References:

  1. Vasil'ev A.N., Tarxov D.A. Nejrosetevoe modelirovanie. Princzipy'. Algoritmy'. Prilozheniya. SPb.: Izd-vo SPbGPU. 2009. 528 s.
  2. Vasil'ev A.N., Tarxov D.A., Osipov V.P. Unificzirovanny'j proczess modelirovaniya fiziko-texnicheskix ob''ektov s raspredelenny'mi parametrami // Nauchno-texnicheskie vedomosti SPbGPU. Fiz.-mat. nauki. 2010. № 3(104). S. 39–52.
  3. Vasil'ev A.N., Tarxov D.A. Nejrosetevy'e podxody' k regulyarizaczii resheniya zadachi prodolzheniya temperaturny'x polej po danny'm tochechny'x izmerenij // Nejrokomp'yutery': razrabotka, primenenie. 2010. № 7. S. 13–19.
  4. Vasil'ev A.N., Porubaev F.V., Tarxov D.A. Nejrosetevoj podxod k resheniyu nekorrektny'x zadach teploperenosa // Nauchno-texnicheskie vedomosti SPbGPU. Informatika. Telekommunikaczii. Upravlenie. 2011. № 1(115). S. 133–142.
  5. Vasil'ev A.N., Tarxov D.A. Postroenie priblizhenny'x nejrosetevy'x modelej po raznorodny'm danny'm // Matematicheskoe modelirovanie. 2007. T. 19. № 12. S. 43–51.
  6. Samarskij A.A., Vabishhevich P.N. Chislenny'e metody' resheniya obratny'x zadach matematicheskoj fiziki. M.: Editorial URSS. 2004. 480 s.
  7. Vasil'ev A.N., Tarxov D.A. Matematicheskie modeli sistem s interval'no zadanny'mi parametrami na osnove geterogenny'x nejronny'x setej. Poristy'j katalizator // Nejrokomp'yutery': razrabotka, primenenie. 2012. № 6.
  8. Vasil'ev A.N., Tarxov D.A. Matematicheskie modeli sistem s interval'no zadanny'mi parametrami na osnove geterogenny'x nejronny'x setej. Prodolzhenie temperaturnogo polya – klassicheskaya postanovka zadachi // Nejrokomp'yutery': razrabotka, primenenie. 2012. № 11. S. 56–59.

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