Abstract
Similarity theory in form of the Pi-Theorem guarantees for any dimensionally homogeneous function the existence of a dimensionless similarity function of dimensionless parameters. Similarity theory is known for its successful applications in science and engineering. Classical applications of similarity theory in engineering mostly exploit the fact that two distinct objects or processes are said to be completely similar if their dimensionless parameters are identical. Similarity theory is used extensively in mathematics for the invariance analysis of differential equations and the derivation of exact and/or approximate solutions. In this work, an extension of the classical applications of similarity theory to artificial intelligence, notably in the fields of case-based and rule-based reasoning, neural networks, data mining, pattern recognition and sound classification, is presented. It is shown that the validity of the extension can be guaranteed by a so-called embedding theorem based on the property of dimensional homogeneity.
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References
Buckingham, E. “On physically similar systems: Illustrations of the use of dimensional equations”, Physical Review, 4, pp. 345–376, 1914.
Bridgman, P. Dimensional Analysis. New Haven, Yale University Press, 1922.
Vaschy, A. “Sur les lois de similitude en Physique”, Annales Télégraphiques (3e série), 19, pp. 25–28, 1892. (French)
Federman, A. “On some general methods of integration of partial differential equations of the first order”, Izvestiya St. Petersburgh Polytechn. Inst, 16, pp. 97–155, 1911. (in Russian)
Riabouchinsky, D.P. “Méthode des variables de dimension zéro, et son application en aérodynamique”, L’Aérophile, 1, Septembre, 407–408, 1911. (in French)
Görtler, H. “Zur Geschichte des Pi-Theorems”, ZAMM, 55, pp. 3–8, 1975. (in German)
Görtler, H. Dimensionsanalyse, Berlin, Springer, 1975. (in German)
Rudolph, S. Übertragung von Ähnlichkeitsbegriffen, Habilitationsschrift, Fakultät für Luft- und Raumfahrttechnik und Geodäsie, Universität Stuttgart, 2002. (in German)
Langhaar, H. Dimensional Analysis and Theory of Models, New York, John Wiley, 1951.
Baker, W., Westine, P. and Dodge, F. Similarity Methods in Engineering Dynamics. Theory and Practice of Scale Modeling. Amsterdam, Elsevier, 1991.
Bluman, G. and Cole, J. Similarity Methods for Differential Equations, New York, Springer, 1974.
Bluman, G. and Kumei, S. Symmetries and Differential Equations. New York, Springer, 1989.
Barenblatt, G. Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge, Cambridge University, Press, 1996.
Kline, S. Similitude and Approximation Theory. Berlin, Springer, 1986.
Huntley, H.Dimensional Analysis. London, MacDonald, 1952.
Duncan, W. Physical Similarity and Dimensional Analysis. London, Arnold, 1953.
Mittelstrass, J. (Hrsg.) Enzyklopädie Philosophie und Wissenschaftstheorie, 4 Bände, Stuttgart, Metzler, 1995. (Satz vom ausgeschlossenen ⇑Widerspruch, in German)
Kolodner, J. Case-Based Reasoning, San Mateo, Morgan Kaufmann, 1993.
Rudolph, S. “On the foundations and applications of similarity theory to case-based reasoning”, Proceedings of the 12th International Conference for Applications of Artificial Intelligence in Engineering (AIENG′97), Capri (Naples), Italy, July 7–9, 1997.
Hertkorn, P. and Rudolph, S. “Dimensional analysis in case-based reasoning”, Proceedings International Workshop on Similarity Methods, University of Stuttgart, Germany, November 26–27, pp. 163–178, 1998.
Sanchez-Sinencio, E. and Lau, C. (eds), Artificial Neural Networks, New York, IEEE Press, 1992.
Hornik, K., Stinchcombe, M. and White, H. “Multilayer feed-forward networks are universal approximators”, Neural Networks, 2, 5, pp. 359–366, 1989.
Rudolph, S. “On topology, size and generalization in non-linear feed-forward neural networks”, Neurocomputing, 16, 1, pp. 1–22, July 1997.
Rudolph, S. “On a data-driven model identification technique using artificial neural networks”, Proceedings of EUROMECH 373 Colloquium on Modelling and Control of Adaptive Mechanical Structures, Magdeburg, Germany, March 11–13, Fortschritt-Berichte VDI, Reihe 11, Nummer 268, pp. 341–350, 1998.
Adriaans, P and Zantinge, D. Data Mining. Harlow, Addison-Wesley, 1996.
Fayyad, U., et al. (eds.) Advances in Knowledge Discovery and Data Mining. Menlo Park, CA: AAAI/MIT Press, 1996.
Hertkorn, P. Knowledge Discovery in Databases auf der Grundlage dimensionshomogener Funktionen, PhD Thesis, Fakultät Luft- und Raumfahrttechnik und Geodäsie, Universität Stuttgart, Stuttgart, 2004. (in German)
Rudolph, S. and Hertkorn, P. “Data mining in scientific data”. In: Data Mining for Design and Manufacturing: Methods and Applications, Braha, D. (ed), pp. 61–85, Dordrecht, Kluwer, 2002.
Brückner, S. and Rudolph, S. “Knowledge discovery in scientific data using hierarchical modeling in dimensional analysis”. Proceedings SPIE Aerosense 2001 Conference On Data Mining and Knowledge Discovery III, Orlando, FL, April 16–20, 2001.
Rudolph, S. “Knowledge discovery in scientific data”, Proceedings SPIE Aerosense Conf. On Data Mining and Knowledge Discovery II, Orlando, FL, ,cm.April 24–28, 2000.
Melan, A. N-dimensionale Merkmalsgewinnung durch vektorielle Dimensionen am Beispiel von Farbbildern und weiteren Anwendungen, PhD Thesis, Fakultät Luft- und Raumfahrttechnik und Geodäsie, Universität Stuttgart, Stuttgart, 2004. (in German)
Till, M. and Rudolph, S. Optimized time-frequency distributions for signal classification with feed-forward neural networks, Proceedings SPIE Aerosense Conference On Applications and Science of Computational Intelligence III, Orlando, FL, April 24–28, 2000.
Till, M. Geräuschklassifikation mittels Übertragung physikalischerÄhnlichkeit, PhD Thesis, Fakultät Luft- und Raumfahrttechnik und Geodäsie, Universität Stuttgart, Stuttgart, 2007. (in German)
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Rudolph, S. (2009). Mathematical Foundations of Non-Classical Extensions of Similarity Theory. In: Borodich, F. (eds) IUTAM Symposium on Scaling in Solid Mechanics. Iutam Bookseries, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9033-2_3
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DOI: https://doi.org/10.1007/978-1-4020-9033-2_3
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