Hyperbolic transfer function

Before training the net, the transfer functions of the neurons must be established. Here, different assays can be made (as detailed in the previous sections), but most often the hyperbolic tangent function tansig function in Table 5.1) is selected for the hidden layer. We set the linear transfer function purelin in Table 5.1) for the output layer. In all cases the output function was the identity function i.e. no further operations were made on the net signal given by the transfer function). 

Using the transfer function concept, Koppel (1967) derived the optimal control policy for a heat exchanger system described by hyperbolic partial differential equations using the lumped system approach. Koppel and Shih (1968) also presented a feedback interior control for a class of hyperbolic differential equations with distributed control. In an earlier paper Koppel e/ al. (1968) discussed the necessary conditions for the system with linear hyperbolic partial differential equations having a control which is independent of spatial coordinates. The optimal feedback-feedforward control law for linear hyperbolic systems, whose dynamical response to input variations is characterized by an initial pure time delay, was derived by Denn... 

The components of the matrix and the vector are called the weights and biases of the layer. Here the hidden layer is denoted as layer 1 and the output layer is denoted as layer 2. Then, the output of the hidden layer is calculated from the transfer function which is chosen to be the hyperbolic tangent sigmoid function in this study ... 

These modified inputs are summed on entry to the neurode. This net input is then modified by an internal transfer function. This might be a step function that passes a signal if a certain threshold is exceeded. More often, a function (such as a sigmoidal or hyperbolic tangent function) that produces a continuous, differentiable nonlinear signal is used (F. 2). The output is passed on either as the input for other neurodes, or as an output carrying a result. 

Using the hyperbolic chart verifying the spatial frequency point where the modulation transfer function drops to 10 % or... 

The hyperbolic tangent (tanh) function, with a sigmoid shape, is the most used transfer function for artificial neural networks. The corresponding kernel has the formula ... 

Fig. 5.16 Hyperbolic tangent sigmoid (a) and linear transfer functions (b)...

The boundary condition at r] = 0 reveals that D = / bubbieCAi and the fact that 4>a vanishes at the outer edge of the mass transfer boundary layer (i.e., at rj = 1) gives C = —D/tanh A. The molar density profile of reactant A in terms of hyperbolic functions is... 

Multiplying the three zone transfer matrices for P22 yields an expression that includes the sum of four products of hyperbolic functions, which is a little long but is easily obtained exactly using computer algebra. In a three-zone reactor, the middle zone typically is the only reactive zone. In the reversible case, Afo = 1 as always and... 

The previous results illustrate how irreversible adsorption leads to the survival of hyperbolic functions in the expressions obtained. This can be avoided by making the reactive zone very thin. If we let L2 0 in the three-zone reactor results, the contribution of the second zone will of course disappear entirely but if we simultaneously increase so that the product k Pz remains fixed and finite, an interesting phenomenon arises the zone transfer matrix tends to the nontrivial result... 

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