Artificial Neural Network (ANN) is a computational model based on the biological neural networks of animal brains. ANN is modeled with three types of layers: an input layer, hidden layers (one or more), and an output layer. Each layer comprises nodes (like biological neurons) are called Artificial Neurons. All nodes are connected with weighted edges (like synapses in biological brains) between two layers. Initially, with the forward propagation function, the output is predicted. Then through backpropagation, the weight and bias to the nodes are updated to minimizing the error in prediction to attain the convergence of cost function in determining the final output.

**AND** logical function truth table for * 2-bit binary variables*, i.e, the input vector and the corresponding output –

0 | 0 | 0 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

Approach:Step1:Import the required Python librariesStep2:Define Activation Function : Sigmoid FunctionStep3:Intialize neural network parameters (weights, bias)

and define model hyperparameters (number of iterations, learning rate)Step4:Forward PropagationStep5:Backward PropagationStep6:Update weight and bias parametersStep7:Train the learning modelStep8:Plot Loss value vs EpochStep9:Test the model performance

**Python Implementation:**

`# import Python Libraries` `import` `numpy as np` `from` `matplotlib ` `import` `pyplot as plt` ` ` `# Sigmoid Function` `def` `sigmoid(z):` ` ` `return` `1` `/` `(` `1` `+` `np.exp(` `-` `z))` ` ` `# Initialization of the neural network parameters` `# Initialized all the weights in the range of between 0 and 1` `# Bias values are initialized to 0` `def` `initializeParameters(inputFeatures, neuronsInHiddenLayers, outputFeatures):` ` ` `W1 ` `=` `np.random.randn(neuronsInHiddenLayers, inputFeatures)` ` ` `W2 ` `=` `np.random.randn(outputFeatures, neuronsInHiddenLayers)` ` ` `b1 ` `=` `np.zeros((neuronsInHiddenLayers, ` `1` `))` ` ` `b2 ` `=` `np.zeros((outputFeatures, ` `1` `))` ` ` ` ` `parameters ` `=` `{` `"W1"` `: W1, ` `"b1"` `: b1,` ` ` `"W2"` `: W2, ` `"b2"` `: b2}` ` ` `return` `parameters` ` ` `# Forward Propagation` `def` `forwardPropagation(X, Y, parameters):` ` ` `m ` `=` `X.shape[` `1` `]` ` ` `W1 ` `=` `parameters[` `"W1"` `]` ` ` `W2 ` `=` `parameters[` `"W2"` `]` ` ` `b1 ` `=` `parameters[` `"b1"` `]` ` ` `b2 ` `=` `parameters[` `"b2"` `]` ` ` ` ` `Z1 ` `=` `np.dot(W1, X) ` `+` `b1` ` ` `A1 ` `=` `sigmoid(Z1)` ` ` `Z2 ` `=` `np.dot(W2, A1) ` `+` `b2` ` ` `A2 ` `=` `sigmoid(Z2)` ` ` ` ` `cache ` `=` `(Z1, A1, W1, b1, Z2, A2, W2, b2)` ` ` `logprobs ` `=` `np.multiply(np.log(A2), Y) ` `+` `np.multiply(np.log(` `1` `-` `A2), (` `1` `-` `Y))` ` ` `cost ` `=` `-` `np.` `sum` `(logprobs) ` `/` `m` ` ` `return` `cost, cache, A2` ` ` `# Backward Propagation` `def` `backwardPropagation(X, Y, cache):` ` ` `m ` `=` `X.shape[` `1` `]` ` ` `(Z1, A1, W1, b1, Z2, A2, W2, b2) ` `=` `cache` ` ` ` ` `dZ2 ` `=` `A2 ` `-` `Y` ` ` `dW2 ` `=` `np.dot(dZ2, A1.T) ` `/` `m` ` ` `db2 ` `=` `np.` `sum` `(dZ2, axis ` `=` `1` `, keepdims ` `=` `True` `)` ` ` ` ` `dA1 ` `=` `np.dot(W2.T, dZ2)` ` ` `dZ1 ` `=` `np.multiply(dA1, A1 ` `*` `(` `1` `-` `A1))` ` ` `dW1 ` `=` `np.dot(dZ1, X.T) ` `/` `m` ` ` `db1 ` `=` `np.` `sum` `(dZ1, axis ` `=` `1` `, keepdims ` `=` `True` `) ` `/` `m` ` ` ` ` `gradients ` `=` `{` `"dZ2"` `: dZ2, ` `"dW2"` `: dW2, ` `"db2"` `: db2,` ` ` `"dZ1"` `: dZ1, ` `"dW1"` `: dW1, ` `"db1"` `: db1}` ` ` `return` `gradients` ` ` `# Updating the weights based on the negative gradients` `def` `updateParameters(parameters, gradients, learningRate):` ` ` `parameters[` `"W1"` `] ` `=` `parameters[` `"W1"` `] ` `-` `learningRate ` `*` `gradients[` `"dW1"` `]` ` ` `parameters[` `"W2"` `] ` `=` `parameters[` `"W2"` `] ` `-` `learningRate ` `*` `gradients[` `"dW2"` `]` ` ` `parameters[` `"b1"` `] ` `=` `parameters[` `"b1"` `] ` `-` `learningRate ` `*` `gradients[` `"db1"` `]` ` ` `parameters[` `"b2"` `] ` `=` `parameters[` `"b2"` `] ` `-` `learningRate ` `*` `gradients[` `"db2"` `]` ` ` `return` `parameters` ` ` `# Model to learn the AND truth table ` `X ` `=` `np.array([[` `0` `, ` `0` `, ` `1` `, ` `1` `], [` `0` `, ` `1` `, ` `0` `, ` `1` `]]) ` `# AND input` `Y ` `=` `np.array([[` `0` `, ` `0` `, ` `0` `, ` `1` `]]) ` `# AND output` ` ` `# Define model parameters` `neuronsInHiddenLayers ` `=` `2` `# number of hidden layer neurons (2)` `inputFeatures ` `=` `X.shape[` `0` `] ` `# number of input features (2)` `outputFeatures ` `=` `Y.shape[` `0` `] ` `# number of output features (1)` `parameters ` `=` `initializeParameters(inputFeatures, neuronsInHiddenLayers, outputFeatures)` `epoch ` `=` `100000` `learningRate ` `=` `0.01` `losses ` `=` `np.zeros((epoch, ` `1` `))` ` ` `for` `i ` `in` `range` `(epoch):` ` ` `losses[i, ` `0` `], cache, A2 ` `=` `forwardPropagation(X, Y, parameters)` ` ` `gradients ` `=` `backwardPropagation(X, Y, cache)` ` ` `parameters ` `=` `updateParameters(parameters, gradients, learningRate)` ` ` `# Evaluating the performance` `plt.figure()` `plt.plot(losses)` `plt.xlabel(` `"EPOCHS"` `)` `plt.ylabel(` `"Loss value"` `)` `plt.show()` ` ` `# Testing` `X ` `=` `np.array([[` `1` `, ` `1` `, ` `0` `, ` `0` `], [` `0` `, ` `1` `, ` `0` `, ` `1` `]]) ` `# AND input` `cost, _, A2 ` `=` `forwardPropagation(X, Y, parameters)` `prediction ` `=` `(A2 > ` `0.5` `) ` `*` `1.0` `# print(A2)` `print` `(prediction)` |

**Output:**

[[ 0. 1. 0. 0.]]

Here, the model predicted output for each of the test inputs are exactly matched with the AND logic gate conventional output () according to the truth table and the cost function is also continuously converging.

Hence, it signifies that the Artificial Neural Network for the AND logic gate is correctly implemented.