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The Hopfield networks that we discussed in the preceding chapter are special recurrent networks, which have a very constrained structure. In this chapter, however, we lift all restrictions and consider recurrent networks without any constraints. Such general recurrent networks are well suited to represent differential equations and to solve them (approximately) in a numerical fashion. If the type of differential equation is known that describes a given system, but the values of the parameters appearing in it are unknown, one may also try to train a suitable recurrent network with the help of example patterns in order to determine the system parameters.
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Due to the special operation scheme of neural networks it is not possible to solve arbitrary differential equations numerically with the help of recurrent networks. It suffices, however, if the differential equation can be solved for the independent variable or for one of the occurring derivatives of the dependent variable. Here we consider, as an example, the special case in which the differential equation can be written with the highest occurring derivative isolated on one side.
A laparoscope is a medical instrument with which a physician can examine the abdominal cavity through small incisions. In virtual laparoscopy an examination of the abdominal cavity is simulated with a computer and a force-feedback device in the shape of a laparoscope in order to teach medical students how to use this instrument properly.
R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics: Mechanics, Radiation, and Heat, vol. 1 (Addison-Wesley, Reading, 1963)
W. Greiner, Teil Mechanik, 1 (Series: Theoretische Physik). Verlag Harri Deutsch, Thun, Frankfurt am Main, Germany, English edition: Classical Mechanics (Springer, Berlin, 1989) (2002)
H. Heuser, Gewöhnliche Differentialgleichungen (Teubner, Stuttgart, 1989)
A. Radetzky, A. Nürnberger, Visualization and simulation techniques for surgical simulators using actual patient’s data. Artif. Intell. Med. 26(3):255–279 (Elsevier Science, Amsterdam, 2002)
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