By Zenon Waszczyszyn (eds.)

Neural Networks are a brand new, interdisciplinary software for info processing. Neurocomputing being effectively brought to structural difficulties that are tough or perhaps most unlikely to be analysed via common pcs (hard computing). The e-book is dedicated to foundations and functions of NNs within the structural mechanics and layout of structures.

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Initial corrvpted 2 .... ... ···· .... ···· ...... .... ••.... ··· ....... •• . .. ••. .. · .... ... .. . ....... .. .. . e. the number of fundamental memories C which can be memorized by Hopfield network composed of N neurons. 138 N . 74) The storage capacity estimation was extensively discussed in literature, cf. [2,17]. 3. 75) 45 Fundamentals of ANNs Function E corresponds to Lyapunov function which is used in the stability analysis of dynamic systems.

The most commonly used RBF is associated with Gaussian functions: Gk N( - (X j = exp L j=l -Ck j2)2j = exp (-[x - C k] TD [x - cd, ) (20') where: D - diagonal weighting matrix D = {Dkk} = {1 /(20'k)} . 49) Fundamentals of ANNs 29 In Fig. 21 b Gaussian function was depicted for x = {x j}, r =IIx -ell, e = 0, a 2 = 2 for} = 1,2. 21c. An application of the classical RBF network is shown to the solution ofXOR problem [2]. 2 for XOR problem. 901 = {I, I }, c 2 = {O, 0 }, and variance (a k )2 = 2. The problem was reduced to the solution of the following equation: G w = t, where: w ={w,w,b}, t = {l,a,I,a} and G is not a square matrix.

Yi(s+l)=Yj(s) forall i. (1. 73) corresponds to the stable state of the network. With the network stable state atractors a r are associated. They are vectors of the input space which attracts the probe vectors presented to the trained network. 34 the attraction of vector x' to a 2 is shown Fig. e. e. with vectors x P from the storage phase as well as with their complements x p . Usually patterns x P are correlated, so also spurious attractors are created. Despite this disadvantage HNNs are frequently used for recognition of corrupted or incomplete patterns.

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