Abstract
This paper builds on advancements in the field of Computational Intelligence to develop a robust approach that combines stochastic optimization methods utilizing Genetic Programming, together with nonlinear evolutionary optimization methods for optimizing the parameters, and Computer Algebra techniques that involve symbolic manipulation of expressions during the course of evolution, to “discover” a parsimonious differential operator that represents an optimum match to the governing differential equation of the target complex nonlinear system and subsequently discloses the correct nature of the investigated system. The proposed scheme requires input and output data only, without postulating any model class in advance. This technique can also discover an accurate single expression, with direct physical interpretation, that represents the governing multi-region (response domain) equations of systems that incorporate certain classes of nonlinear phenomena (such as yielding). Yet, unlike many conventional nonparametric techniques whose approximations result in undesirable oscillations around unsmooth points, automatic incorporation of discontinuous basis functions in this approach eliminates the need of such approximations and their concomitant errors. A variety of highly nonlinear phenomena are considered to assess the capabilities and the generalization extent of the suggested approach. It is shown that the method of this paper provides a robust methodology for develo** reduced-order, reduced-complexity, computational models (in the form of governing differential equations) that can be used for obtaining high-fidelity models that reflect the correct “physics” of the underlying phenomena.
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References
Ibanez, P.: Identification of dynamic parameters of linear and non-linear structural models from experimental data. Nucl. Eng. Des. 25(1), 30–41 (1973)
Masri, S.F., Caughey, T.K.: A nonparametric identification technique for nonlinear dynamic problems. J. Appl. Mech. 46(2), 433–447 (1979)
Masri, S.F., Sassi, H., Caughey, T.K.: Nonparametric identification of nearly arbitrary nonlinear systems. J. Appl. Mech. 49(3), 619–628 (1982)
Marsi, S.F., Bekey, G.A., Sassi, H., Caughey, T.K.: Non-parametric identification of a class of non-linear multidegree dynamic systems. Earthq. Eng. Struct. Dyn. 10(1), 1–30 (1982)
Yun, H.B., Tasbighoo, F., Masri, S.F., Caffrey, J.P., Wolfe, R.W., Makris, N., Black, C.: Comparison of modeling approaches for full-scale nonlinear viscous dampers. J. Vib. Control 14(1–2), 51–76 (2008)
Yun, H.B., Masri, S.F., Wolfe, R.W., Benzoni, G.: Data-driven methodologies for change detection in large-scale nonlinear dampers with noisy measurements. J. Sound Vib. 322(1–2), 336–357 (2009)
Kim, C.H., Mijar, A.R., Arora, J.S.: Development of simplified models for design and optimization of automotive structures for crashworthiness. Struct. Multidiscip. Optim. 22(4), 307–321 (2001)
Allen, M.S., Sumali, H., Epp, D.S.: Piecewise-linear restoring force surfaces for semi-nonparametric identification of nonlinear systems. Nonlinear Dyn. 54(1–2), 123–135 (2008)
Crawley, E.F., Aubert, A.C.: Identification of nonlinear structural elements by force-state map**. AIAA j. 24(1), 155–162 (1986)
Benedettini, F., Capecchi, D., Vestroni, F.: Identification of hysteretic oscillators under earthquake loading by nonparametric models. J. Eng. Mech. 121(5), 606–612 (1995)
Shin, K., Hammond, J.K.: Pseudo force-state map** method: incorporation of the embedding and force-state map** methods. J. Sound Vib. 211(5), 918–922 (1998)
Haroon, M., Douglas, E.A., Luk, Y.W., Ferri, A.A.: A time and frequency domain approach for identifying nonlinear mechanical system models in the absence of an input measurement. J. Sound Vib. 283(3–5), 1137–1155 (2005)
Ajjan Al-Hadid, M., Wright, J.R.: Developments in the force-state map** technique for non-linear systems and the extension to the location of non-linear elements in a lumped-parameter system. Mech. Syst. Signal Process. 3(3), 269–290 (1989)
Box, G.: Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco (1970)
Leontaritis, I.J., Billings, S.A.: Input-output parametric models for non-linear systems. part i: deterministic non-linear systems. Int. J. Control 41(2), 303–328 (1985)
Leontaritis, I.J., Billings, S.A.: Input-output parametric models for non-linear systems. Part ii: stochastic non-linear systems. Int. J. Control 41(2), 329–344 (1985)
Haitao, L., Mita, A.: Damage indicator defined as the distance between arma models for structural health monitoring. Struct. Control Health Monit. 15(7), 992–1005 (2008)
Kerschen, G., Worden, K., Vakakis, A.F., Golinval, J.-C.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20(3), 505–592 (2006)
Saito, T., Beck, J.L.: Bayesian model selection for arx models and its application to structural health monitoring. Earthq. Eng. Struct. Dyn. 39(15), 1737–1759 (2010)
Masri, S.F., Chassiakos, A.G., Caughey, T.K.: Structure-unknown non-linear dynamic systems: identification through neural networks. Smart Mater. Struct. 1(1), 45–56 (1992)
Masri, S.F., Chassiakos, A.G., Caughey, T.K.: Identification of nonlinear dynamic systems using neural networks. Trans. ASME J. Appl. Mech. 60(1), 123–133 (1993)
Worden, K., Tomlinson, G.R.: Modeling and classification of non-linear systems using neural networks-i. simulation. Mech. Syst. Signal Process. 8(3), 319–356 (1994)
Worden, K., Tomlinson, G.R., Lim, W., Sauer, G.: Modeling and classification of non-linear systems using neural networks. ii. a preliminary experiment. Mech. Syst. Signal Process. 8(4), 395–419 (1994)
Chassiakos, A.G., Masri, S.F.: Modelling unknown structural systems through the use of neural networks. Earthq. Eng. Struct. Dyn. 25(2), 117–128 (1996)
Smyth, A.W., Masri, S.F., Chassiakos, A.G., Caughey, T.K.: On-line parametric identification of mdof nonlinear hysteretic systems. J. Eng. Mech. 125(2), 133–142 (1999)
Masri, S.F., Smyth, A.W., Chassiakos, A.G., Caughey, T.K., Hunter, N.F.: Application of neural networks for detection of changes in nonlinear systems. J. Eng. Mech. 126(7), 666–676 (2000)
Pei, J.S., Smyth, A.W.: New approach to designing multilayer feedforward neural network architecture for modeling nonlinear restoring forces. i: formulation. J. Eng. Mech. 132(12), 1290–1300 (2006)
Pei, J.S., Smyth, A.W.: New approach to designing multilayer feedforward neural network architecture for modeling nonlinear restoring forces ii: applications. J. Eng. Mech. 132(12), 1301–1312 (2006)
Darwin, C.: The Origin of Species by Means of Natural Selection or the Preservation of Favored Races in the Struggle for Life. Murray, London (1859)
Koza, J.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge (1992)
Tackett, W.A.: Genetic Programming for Feature Discovery and Image Discrimination. San Mateo, CA (1993)
Gruau, F.: Automatic definition of modular neural networks. Adapt. Behav. 3(2), 151–183 (1994)
Spector, L., Alpern, A.: Induction and Recapitulation of Deep Musical Structure. Montreal, Quebec (1995)
Handley, S.: Predicting Whether or Not a Nucleic Acid Sequence is an e. coli Promoter Region using Genetic Programming. Herndon, VA (1995)
Wong, M.L., Leung, K.S., Cheng, J.C.Y.: Discovering knowledge from noisy databases using genetic programming. J. Am. Soc. Inf. Sci. 51(9), 870–881 (2000)
Howard, D., Roberts,S.C.: The Prediction of Journey Times on Motorways using Genetic Programming, pp. 210–21. Berlin, Germany (2002)
Becker, Y., Fei, P., Lester, A.: Stock selection: an innovative application of genetic programming methodology. In: Riolo, R., Soule, T., Worzel, B. (eds.) Genetic Programming Theory and Practice IV, Genetic and Evolutionary Computation, pp. 315–334. Springer, US (2007)
Schmidt, M., Lipson, H.: Distilling free-form natural laws from experimental data. Science 324(5923), 81–85 (2009)
Hornby, G.S., Lohn, J.D., Linden, D.S.: Computer-automated evolution of an x-band antenna for nasa’s space technology 5 mission. Evolut. Comput. 19(1), 1–23 (2011)
Silva, S., Anunciação, O., Lotz, M.: A comparison of machine learning methods for the prediction of breast cancer. In: Evolutionary Computation, Machine Learning and Data Mining in Bioinformatics, pp. 159–170. Springer, Berlin (2011)
Dracopoulos, D., Effraimidis, D.: Genetic programming for generalised helicopter hovering control. In: Genetic Programming, volume 7244 of Lecture Notes in Computer Science, pp. 25–36. Springer, Berlin (2012)
Fisher, R.A.: The Genetical Theory of Natural Selection. Diver, NY (1958)
Fogel, L.J., Owens, A.J., Walsh, M.J.: Artificial Intelligence Through Simulated Evolution. Wiley, New York (1966)
Schwefel, H.P.: Kybernetische evolution als strategie der experimentellen forschung in der stromungstechnik (1973)
Rechenberg, I.: Evolutionsstrategie: Optimierung technischer systeme nach prinzipien der biologischen evolution. Frommann-Holzboog (1973)
Holland, J.H.: Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor (1975)
Thierens, D., Goldberg, D.: Convergence Models of Genetic Algorithm Selection Schemes, pp. 119–29. Berlin, Germany (1994)
Fleming, P.J., Purshouse, R.C.: Evolutionary algorithms in control systems engineering: a survey. Control Eng. Pract. 10(11), 1223–1241 (2002)
Baker, J.E.: Reducing Bias and Inefficiency in the Selection Algorithm, pp. 14–21. Hillsdale, NJ (1987)
Goldberg, D.E., Deb, K.: A Comparative Analysis of Selection Schemes used in Genetic Algorithms, pp. 69–93. Foundations of genetic algorithms (1991)
Langdon, W.B., Poli, R.: Genetic programming bloat with dynamic fitness. In: Banzhaf, W., Poli, R., Schoenauer, M., Fogarty, T.C. (eds.) The First European Workshop on Genetic Programming, volume 1391 of LNCS, pp. 96112. Springer, Paris (1998)
Banzhaf, W., Langdon, W.B.: Some considerations on the reason for bloat. Genet. Program. Evolvable Mach. 3(1), 81–91 (2002)
Silva, S., Costa, E.: Dynamic Limits for Bloat Control: Variations on Size and Depth. pp. 666–77. Berlin, Germany (2004)
Luke, S., Panait, L.: Lexicographic parsimony pressure. In: Langdon, W.B., et al. (eds.) GECCO 2002: Proceedings of the Genetic and Evolutionary Computation Conference, pp. 829–836. Morgan Kaufmann Publishers, New York (2002)
Gagne, C., Schoenauer, M., Parizeau, M., Tomassini, M.: Genetic Programming, Validation Sets, and Parsimony Pressure, pp. 109–20. Berlin, Germany (2006)
Masri, S.F., Caffrey, J.P., Caughey, T.K., Smyth, A.W., Chassiakos, A.G.: Identification of the state equation in complex non-linear systems. Int. J. Non-Linear Mech. 39(7), 1111–1127 (2004)
Baker, G.A., Graves-Morris, P.: Pade Approximations, 2nd edn. Cambridge University Press, Cambridge (1996)
Emmel, L., Kaber, S.M., Maday, Y.: Pade-jacobi filtering for spectral approximations of discontinuous solutions. Numer. Algorithms 33(1–4), 251–264 (2003)
Kaber, S.M., Maday, Y.: Analysis of some pade-chebyshev approximants. SIAM J. Numer. Anal. 43(1), 437–454 (2005)
Hesthaven, J.S., Kaber, S.M., Lurati, L.: Pade-legendre interpolants for gibbs reconstruction. J. Sci. Comput. 28(2–3), 337–359 (2006)
Foster, J., Richards, F.B.: The gibbs phenomenon for piecewise-linear approximation. Am. Math. Mon. 98(1), 47–49 (1991)
Driscoll, T.A., Fornberg, B.: A pade-based algorithm for overcoming the gibbs phenomenon. Numer. Algorithms 26(1), 77–92 (2001)
Shizgal, B.D., Jung, J.H.: Towards the resolution of the gibbs phenomena. J. Comput. Appl. Math. 161(1), 41–65 (2003)
Chantrasmi, T., Doostan, A., Iaccarino, G.: Pade-legendre approximants for uncertainty analysis with discontinuous response surfaces. J. Comput. Phys. 228(19), 7159–7180 (2009)
Lew, T.L., Spencer, A.B., Scarpa, F., Worden, K., Rutherford, A., Hemez, F.: Identification of response surface models using genetic programming. Mech. Syst. Signal Process. 20(8), 1819–1831 (2006)
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The authors acknowledge the financial support by King Abdulaziz City for Science and Technology (KACST) through contract number 32-710. The support of Prince Dr. Turki Bin Saud Al-Saud KACST VP is appreciated.
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Bolourchi, A., Masri, S.F. & Aldraihem, O.J. Development and application of computational intelligence approaches for the identification of complex nonlinear systems. Nonlinear Dyn 79, 765–786 (2015). https://doi.org/10.1007/s11071-014-1702-9
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DOI: https://doi.org/10.1007/s11071-014-1702-9