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Figure 2.14 Energy levels in a GaAs single quantum well with constant effective mass m*=0.067m0 and F=100 meV

is calculated as measure of how well the NN approximates the true function. Another very important aspect of NN accuracy is overfitting. Overfitting of a training set means that the NN memorizes the training patterns, and consequently loses the ability to generalize. That is, NNs that overfit cannot predict correct output

introduce r-r' and d-d' by 'subtracting' the bottom line from the top }

methods' [3,7,18]. While it is interesting to view the functional form of the equations, such methods are time consuming and inefficient and wouldn't be employed in the repetitive solution of many quantum wells. Computationally it is much more effective just to treat equations (2.69) and (2.70) with standard techniques, such as NewtonRaphson iteration. In this technique, if E(n)isafirst guess to the solution of f ( E ) 0, then a better estimate is given by:

- Q < r - r ' < Q A 0 = Qx(d-d') + (r-r') { { { arithmetic and Leibniz } 0 < Q, inequalities } A -(Q.x(d-d')) = r-r' (integer) arithmetic and Leibniz } -Q < -(Qx(d-d')) < Q A -(Qx(d-d')) =r-r' -l<-(d-d')<l

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The new estimate *( +!) is then used to generate a second approximation to the solution E(n+2) and so on, until the successive estimates converge to a required accuracy. In order to provide all of the required information to implement the solution, all that remains is to deduce f ' ( E ) . For the even parity states:

LEARNING)

d = d' A r = r' .

and so therefore:

In this section, we develop an elementary algorithm for computing remainders and the corresponding quotients. The algorithm is not efficient, but its existence allows us to identify a number of basic algebraic properties of remainders. The algorithm involves a case analysis on positive and negative numbers P. The case that P is positive is developed in detail below. Exercise 15.5 asks you to develop the algorithm in the case that P is negative. The elementary algorithm is based on splitting (15.1) into P = Qxd + r (15.3) (15.4)

For the odd-parity states:

for data patterns not seen during training. Overfitting occurs when the NN architecture is too large, i.e. the NN has too many weights (in statistical terms: too many free parameters) - a direct consequence of having too many hidden units and irrelevant input units. If the NN is trained for too long, the excess free parameters start to memorize all the training patterns, and even noise contained in the training set. Remedies for overfitting include optimizing the network architecture and using enough training patterns (discussed in Section 7.3). Estimations of generalization error during training can be used to detect the point of overfitting. The simplest approach to find the point of overfitting was developed through studies of training and generalization profiles. Figure 7.1 presents a general illustration of training and generalization errors as a function of training epochs. From the start of training, both the training and generalization errors decrease - usually exponentially. In the case of oversized NNs, there is a point at which the training error continues to decrease, while the generalization error starts to increase. This is the point of overfitting. Training should stop as soon an increase in generalization error is observed. Error

r,d := P,0 .

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