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International Conference on Artificial Neural Networks

ICANN 2005: Artificial Neural Networks: Formal Models and Their Applications – ICANN 2005 pp 391–396Cite as

Topological Derivative and Training Neural Networks for Inverse Problems

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3697))

Abstract

We consider the problem of locating small openings inside the domain of definition of elliptic equation using as the observation data the values of finite number of integral functionals. Application of neural networks requires a great number of training sets. The approximation of these functionals by means of topological derivative allows to generate training data very quickly. The results of computations for 2D examples show, that the method allows to determine an approximation of the global solution to the inverse problem, sufficiently closed to the exact solution.

Supported by the grant 4-T11A-015-24 of the State Committee for the Scientific Research of the Republic of Poland.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/11550907_163 .

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References

  1. Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory 39, 930–945 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Hagan, M., Menhaj, M.: Training feedforward networks with the Marquardt algorithm. IEEE Trans. on Neural Networks 5, 989–993 (1994)

    Article  Google Scholar 

  3. Il’in, A.M.: Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. In: Translations of Mathematical Monographs, vol. 102. AMS, Providence (1992)

    Google Scholar 

  4. Jackowska, L., Sokołowski, J., Żochowski, A., Henrot, A.: On numerical solutions of shape inverse problems. Computational Optimization and Applications 23, 231–255 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lewiński, T., Sokołowski, J., Żochowski, A.: Justification of the bubble method for the compliance minimization problems of plates and spherical shells CD-Rom. In: 3rd World Congress of Structural and Multidisciplinary Optimization (WCSMO-3) Buffalo/Niagara Falls, New York, May 17-21 (1999)

    Google Scholar 

  6. Nazarov, S.A., Sokołowski, J.: Asymptotic analysis of shape functionals, Les prépublications de l’Institut Élie Cartan, 51 (2001)

    Google Scholar 

  7. Roche, J.R., Sokołowski, J.: Numerical methods for shape identification problems. Special issue of Control and Cybernetics: Shape Optimization and Scientific Computations 5, 867–894 (1996)

    Google Scholar 

  8. Schiffer, M., Szegö, G.: Virtual mass and polarization. Transactions of the American Mathematical Society 67, 130–205 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  9. Shumacher, A.: Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungkriterien, Ph.D. Thesis, Universität–Gesamthochschule–Siegen, Siegen (1995)

    Google Scholar 

  10. Sokołowski, J., Żochowski, A.: On topological derivative in shape optimization. SIAM Journal on Control and Optimization 37(4), 1251–1272 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Sokołowski, J., Żochowski, A.: Topological derivative for optimal control problems. Control and Cybernetics 28(3), 611–626 (1999)

    MATH  MathSciNet  Google Scholar 

  12. Sokołowski, J., Żochowski, A.: Topological derivatives for elliptic problems. Inverse Problems 15(1), 123–134 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sokołowski, J., Żochowski, A.: Topological derivatives of shape functionals for elasticity systems. Mechanics of Structures and Machines 29, 333–351 (2001)

    Google Scholar 

  14. Sokołowski, J., Żochowski, A.: Optimality conditions for simultaneous topology and shape optimization. SIAM Journal on Control and Optimization 42(4), 1198–1221 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Computer Engineering Department, Technical University of Łódz, Al. Politechniki 11, 90-924, Łódz, Poland

    Lidia Jackowska-Strumiłło

  2. Institut Elie Cartan, Université Henri Poincaré, Nancy I, B.P. 239, 54506, Vandoeuvre lès Nancy Cedex, France

    Jan Sokołowski

  3. Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447, Warszawa, Poland

    Antoni Żochowski

Authors
  1. Lidia Jackowska-Strumiłło
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  2. Jan Sokołowski
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  3. Antoni Żochowski
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Editor information

Editors and Affiliations

  1. Department of Informatics, Nicolaus Copernicus University, Toruń, Poland

    Włodzisław Duch

  2. Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01–447, Warsaw, Poland

    Janusz Kacprzyk

  3. Adaptive Informatics Research Centre, Helsinki University of Technology, P.O. Box 5400, 02015, HUT, Finland

    Erkki Oja

  4. Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Sławomir Zadrożny

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© 2005 Springer-Verlag Berlin Heidelberg

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Jackowska-Strumiłło, L., Sokołowski, J., Żochowski, A. (2005). Topological Derivative and Training Neural Networks for Inverse Problems. In: Duch, W., Kacprzyk, J., Oja, E., Zadrożny, S. (eds) Artificial Neural Networks: Formal Models and Their Applications – ICANN 2005. ICANN 2005. Lecture Notes in Computer Science, vol 3697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11550907_62

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  • DOI: https://doi.org/10.1007/11550907_62

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28755-1

  • Online ISBN: 978-3-540-28756-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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