1. Artificial Neural Network

2. Definition
• Artificial Neural network or ANN is a very
popular method for predictive or optimization
or simulation objectives.
• ANN mimics the human nervous system to
solve problems in a parallel manner.
• ANN are known to be adaptable with
situations, flexible with data and efficient
enough for predicting any kind of problems.

3. Mathematical Representation of ANN
Single Layer Neural Network Multi Layer Neural Network

4. Mathematical Representation of ANN
Single Layer Neural Network
Multi Layer Neural Network
Yj = gA ( ∑hij x Xi + aj )
Where Y ,Z are the output hj are the hidden nodes and Xi are the inputs, f and g are the
activation function for the hidden to output and input to hidden layers respectively, Xi are
the inputs, w and h are the weights of respectively for hidden to output and input to
hidden layer connections and a,b are the bias. i = 1…n,j = 1…h.
Zk = fA ( ∑wjk x hj + bk)
and
hj = gA (∑ hij x Xi + aj )
Eqn 1
Eqn 2
Eqn A

5. Basic Methodology of ANN
1. Selection of the Model Topology : ANN have an input and output
layer. Between this two layers, lies the hidden layer which actually
separates the ANN model from the other linear and non-linear
models. Selection of the number of hidden layers influences the
efficiency of the model. More the number of hidden layers more
complex but efficient will be the model and vice versa.
2. Training for determination of the optimal value of the weights.
The weight of the inputs are changed to equate the predicted
value with the desired value of the output. Whenever both the
desired and predicted value becomes equal or nearly equal to the
satisfaction of the developer the training is stopped.
3. Validation of the Model by predicting the known outputs.

6. Problem 1
• A two input-one output model is required to
be developed. The training data for the model
is as follows :
• Input 1 : 10
• Input 2 : 12
• Output 1 : 24.
• Find the value of the output when Input 1 is 5
and Input 2 is 7 if number of hidden layer is 1
and node is 2.

7. Solution
• The architecture of the ANN will be as below :
Input 1
Input 2
Hidden 1
Hidden 2
Output 1
h 11
h 12
h 21
h 22
w 11
w 21

8. Solution
• As the ANN will be multilayer : Input, Hidden
and Output layers we will use Eqn 1 and Eqn 2
to find the optimal weights first.
• Then we will use the optimal weights in the
same equation to find the output or answer of
the problem

9. Solution
• Zk = fA ( ∑wjk x hj + bk)
And
• hj = gA (∑ hij x Xi + aj )
Here X1 and X2 is 10 and 12 respectively
And Z1 is 24
Let hij the weights be 0.5 and wjk be 05.

10. Solution
• Replacing we have :
• Eqn 2 =
• h1 = ga(h11X10+h21X12)+ a1)
• if ga = Logarithmic function(aLog(X)) and a1 is
negligible then taking h11 and h22 as 0.5 will give
the following :
• 10xLOG (0.5x10+0.5x12) = 10xLOG(5+6) =
10xLOG(11) =10.41 = h1
• Similarly h2 will also be 10.41

11. Solution
• As, Zk = fA ( ∑wjk x hj + bk)
• Then we can write :
• Eqn 1 =
• Z1 = fA ( 0.5xh1 + 0.5xh2) + b1 )
• If fA is taken as Logarithmic and we know that h1 =
h2 = 10.41
• then
• Z1 = 10xLog(0.5x10.41+0.5x10.41)
• = 10.17 = Eqn.1

12. Solution
• Now the desired or given output is 24
• So Absolute Error or AE = 24-10.17 =13.82
• So we have to change the value of h and w
and recalculate the output.
• When the AE is less than 20% the value of
weights (h and w) will be taken as optimal.
• The same Eqn 1 and 2 will be used. This time
we know the value of inputs and weights but
do not know the value of output.

13. Solution
• Eqn 2 =
• h1 = ga(h11X5+h21X7)+ a1)
• if ga = Logarithmic function and a1 is negligible
and optimal value h11 and h22 as 1.5 will give
the following :
• 10xLOG (1.5x5+1.5x7) = 12.55 = h1
• Similarly h2 will also be 12.55

14. Solution
• As, Zk = fA ( ∑wjk x hj + bk)
• Then we can write :
• Eqn 1 =
• Z1 = fA ( 1.5xh1 + 1.5xh2) + b1 )
• If fA is taken as Logarithmic and we know that h1 =
h2 = 12.55
• then
• Z1 = 10xLog(1.5x12.55+1.5x12.55)
• = 15.75 = Eqn.1
• So the output or answer is 15.75.