solve this---**Title:** Back Propagation in Artificial Neural Networks – Solved Example
**Problem Description:**
* Consider a multilayer feed-forward neural network as shown in the figure.
* Let the learning rate be 0.5.
* Train the network for the training tuple (1, 1, 0) where last number is target output.
* Show weight and bias updates by using back-propagation algorithm.
**Neural Network Diagram Description:**
* **Type:** Feed-forward neural network diagram.
* **Structure:** The network has an input layer, a hidden layer, and an output layer.
* **Nodes:**
* Input Layer: Two nodes labeled X1 and X2, represented by squares.
* Hidden Layer: Two nodes labeled 3 and 4, represented by circles.
* Output Layer: One node labeled 5, represented by a circle.
* **Connections and Weights:** Directed lines (arrows) connect nodes between layers, indicating the flow of information. Each connection has an associated weight.
* From X1 to 3: Weight W13 = 0.5
* From X1 to 4: Weight W14 = 0.2
* From X2 to 3: Weight W23 = -0.3
* From X2 to 4: Weight W24 = 0.5
* From 3 to 5: Weight W35 = 0.1
* From 4 to 5: Weight W45 = 0.3
* **Biases:** Each node in the hidden and output layers has an associated bias, indicated by an incoming arrow pointing towards the node from a non-connected source.
* Bias for node 3: b3 = 0.6
* Bias for node 4: b4 = -0.4
* Bias for node 5: b5 = 0.8
* **Flow:** Arrows show the direction of signal propagation from input (X1, X2) to hidden layer (3, 4) and then to the output layer (5).
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We have a three-layer neural network with two input nodes X1 and X2, two hidden nodes labeled 3 and 4, and one output node labeled 5. The network has initial weights and biases as shown. We need to train this network using backpropagation with the training tuple where inputs are 1 and 1, target output is 0, and learning rate is 0.5.
Now we perform forward propagation. With inputs X1 equals 1 and X2 equals 1, we calculate the net inputs and outputs for each layer. For hidden node 3, net input is 0.8 giving output 0.689. For hidden node 4, net input is 0.3 giving output 0.574. The final output is 0.739, but our target is 0, so we have an error.
In step 2, we calculate the error and gradients. The total error is 0.273. For the output layer, we compute delta 5 as negative 0.142. This gives us the gradients for the output layer weights and bias, which we'll use to update the parameters.
In step 3, we calculate hidden layer gradients using backpropagation. Delta 3 is negative 0.003 and delta 4 is negative 0.010. In step 4, we update all weights using learning rate 0.5. The output layer weights increase slightly, while hidden layer weights show minimal changes due to small gradients.
To summarize, we successfully demonstrated backpropagation on a three-layer neural network. The algorithm computed gradients layer by layer, updated all weights and biases, and moved the network closer to the target output. This iterative process would continue until the network achieves acceptable accuracy.
Now we perform forward propagation. With inputs X1 equals 1 and X2 equals 1, we calculate the net inputs and outputs for each layer. For hidden node 3, net input is 0.8 giving output 0.689. For hidden node 4, net input is 0.3 giving output 0.574. The final output is 0.739, but our target is 0, so we have an error.
In step 2, we calculate the error and gradients. The total error is 0.273. For the output layer, we compute delta 5 as negative 0.142. This gives us the gradients for the output layer weights and bias, which we'll use to update the parameters.
In step 3, we calculate hidden layer gradients using backpropagation. Delta 3 is negative 0.003 and delta 4 is negative 0.010. In step 4, we update all weights using learning rate 0.5. The output layer weights increase slightly, while hidden layer weights show minimal changes due to small gradients.
To summarize, we successfully demonstrated backpropagation on a three-layer neural network. The algorithm computed gradients layer by layer, updated all weights and biases, and moved the network closer to the target output. This iterative process would continue until the network achieves acceptable accuracy.