Download e-book for kindle: Analyse convexe et problemes variationnels by I Ekeland

By I Ekeland

ISBN-10: 204007368X

ISBN-13: 9782040073688

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The basic model of the neuron is founded upon the functionality of a biological neuron. “Neurons are the basic signaling units of the nervous system” and “each neuron is a discrete cell whose several processes arise from its cell body”. The neuron has four main regions to its structure. The cell body, or soma, has two offshoots from it, the dendrites, and the axon, which end in presynaptic terminals. The cell body is the heart of the cell, containing the nucleus and maintaining protein synthesis.

26 3 Preliminaries of Neural Networks Fig. 3 Radial Basis Function Neural Network Radial basis function (RBF) network consists of three layers, input layer is first layer and basis function is the second layer as hidden layer and an output layer as shown in Fig. 10. Each node in the hidden layer represents a Gaussian basis k be function for all nodes and output node uses a linear activation function. Let WRBF the vector connection weight between the input nodes and the k-th RBF node or we k ¼ X À W k ; so the output of the k-th RBF node is can say WRBF   1 k  2 hkRBF ¼ exp À 2 WRBF rk ð3:12Þ where rk is the spread of k-th RBF function, X ¼ ðx1 ; x2 ; .

For a given input vector ~ x ¼ ðx1 ; x2 ; . ; xn Þ the output of the network can be given as: N¼ H X vi rðzi Þ ð4:5Þ i¼1 where, zi ¼ n X wij xj þ ui ; j¼1 In Eq. 5) wij denotes the weight from the input unit j to the hidden unit i, vi represents weight from the hidden unit i to the output, ui is the bias of hidden unit i, and rðzÞ is the sigmoid activation function. Now the derivative of networks output N with respect to input vector xj is: @N @ ¼ @xj @xj H X i¼1 vi r n X !! wij xj þ ui j¼1 ¼ h X vi wij rð1Þ ð4:6Þ i¼1 where, rð1Þ ¼ @rð xÞ @x Similarly, the k-th derivative of N is @k N X ðk Þ ¼ vi wkij ri @xkj ð4:7Þ where, ri ¼ rðzi Þ and rðkÞ denotes the k-th order derivative of the sigmoid activation function.

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Analyse convexe et problemes variationnels by I Ekeland


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