Optimization

Optimization

Feedback Neural Network Training Mode

Optimization

Once the inverse network is trained, the parameters need to be optimized. This process is referred to as the training mode. The goal of the optimization procedure is to minimize the error due to the inverse radial basis function network. Let Df be the error between the actual MFL signal and the prediction of the forward network in the feedback configuration. In order for Df to be zero, the characterization network must be an exact inverse of the forward network. While the functional form of the forward network can be derived easily, obtaining its inverse analytically is difficult. This is because the output of the forward network is a function of the number and location of their respective basis function centers in each network. The inverse is, therefore, estimated numerically.

Inverse Neural Network Optimization

An adaptive scheme is used to estimate the inverse of the forward network as shown above. This "inverse network" is used as the characterization network. The algorithm uses gradient descent in combination with simulated annealing to optimize the inverse network parameters. The algorithm is shown at right, and the derivation of the network parameter update equations is shown below. As shown in the flowchart, the defect profile is input to the forward network. The output of the forward network (the predicted signal) is input to the inverse network. The defect profile predicted by the inverse network is compared to the true profile and the error is used to update the inverse network parameters. This occurs only if the error after the update is less than the error before the update. In case the error increases, the update is retained with a probability p that decreases as the number of iterations increases. This probabilistic update rule is used to escape from local minima, which the gradient descent rule is susceptible to get trapped in.

Let E = the error at the output of the inverse network,

w_kj = interconnection weight from node j in the hidden layer to node k in the output layer

c_j = center of the j^th basis function (at node j in the hidden layer)

s_j = spread of the j^th basis function

f = the signal

be the desired output of the radial basis function network

be the actual output of the radial basis function network

Then, the error E can be defined as

^{where ^{is given by}}

and the basis function is chosen to be a Gaussian function:

Substituting the second two equations into the first equation and taking the derivative with respect to the weights w_kj, we have

Similarly, the derivative of the error with respect to the other two parameters (c_j and s_j) can be computed as follows:

The derivatives are then substituted into the gradient descent equation to derive the update equations for the three parameters. These expressions are given by the following equations.

Once the characterization network is trained and optimized, the two networks are connected in the feedback configuration shown earlier. The characterization network can then be used for predicting flaw profiles using signals obtained from defects of unknown shape and size.

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