Forward propagation is the process of taking the input data, passing it through the network's layers, and computing an output prediction. It's the "prediction phase" of the neural network.
Intuition:
Imagine a factory assembly line.
1. Inputs are the raw materials.
2. Each neuron/layer is a processing station. At each station:
- Materials from the previous station arrive (inputs).
- Each material piece is weighted (multiplied by a weight).
- All weighted materials are combined (summed up), and an additional adjustment is made (bias).
- This combined material then goes through a "quality control" filter (activation function) that transforms it.
- The transformed material is then passed on to the next station.
3. This process continues, layer by layer, until the final product (the prediction) emerges at the output layer.
Step-by-Step Flow:
1. Input Layer: The raw input features are fed into the input layer.
2. First Hidden Layer:
- For each neuron in the first hidden layer:
- It receives input values from all neurons in the input layer.
- Each input value (x_i) is multiplied by its corresponding weight (w_i).
- These weighted inputs are summed up: sum(x_i * w_i).
- A bias term (b) is added to this sum: Z = sum(x_i * w_i) + b.
- The sum Z is then passed through an activation function (e.g., ReLU, Sigmoid) to produce the neuron's output (activation value).
- These activation values become the inputs for the next layer.
3. Subsequent Hidden Layers (if any): The same process from step 2 is repeated for each subsequent hidden layer. The outputs (activations) of the previous hidden layer serve as inputs for the current hidden layer.
4. Output Layer:
- The final hidden layer's activations are fed into the output layer.
- Similar weighted sum and bias calculations are performed.
- A final activation function (e.g., Sigmoid for binary classification, Softmax for multi-class classification, or linear for regression) is applied to produce the network's final prediction(s).
At the end of forward propagation, the network has made a prediction based on its current set of weights and biases.

Backpropagation is the algorithm that enables the neural network to learn. It's the process of calculating how much each weight and bias in the network contributed to the error in the final prediction, and then using this information to adjust those weights and biases to reduce future errors. It's essentially the "learning phase."
Intuition: The Credit Assignment Problem
Imagine the factory assembly line again.
1. Error Detection: At the very end of the line, the final product (prediction) is inspected, and an error is found (the difference between the prediction and the actual desired output).
2. Blame Assignment (Backward Pass): Instead of just complaining about the final product, backpropagation works backward through the assembly line.
- It starts by determining which part of the last station's processing (output layer) contributed most to the error. It calculates how much each weight and bias in that last layer needs to change to reduce the error.
- Then, it passes this "blame" or "gradient" backward to the previous station (hidden layer). It figures out how much each output from that previous station contributed to the error at the current station, and consequently, how much the weights and biases of that previous station need to be adjusted.
- This "blame" is propagated backward, layer by layer, through the entire network, calculating the contribution of every single weight and bias to the overall error.
3. Weight Adjustment (Optimization): Once the "blame" (gradient) is known for every weight and bias, an optimizer uses this information to make small adjustments to all weights and biases. The adjustments are made in a direction that is expected to reduce the error.
Step-by-Step Flow (Intuitive):
1. Calculate Loss: After forward propagation, the network's prediction is compared to the true (actual) value using a loss function (e.g., Mean Squared Error for regression, Cross-Entropy for classification). This calculates a single numerical value representing the network's error.
2. Calculate Gradients for Output Layer: The backpropagation algorithm starts by calculating the gradient of the loss with respect to the weights and biases of the output layer. This tells us how much the loss would change if we slightly adjusted each weight or bias in the output layer.
3. Propagate Gradients Backward: Using the chain rule of calculus, these gradients are then propagated backward through the network, layer by layer, to the hidden layers. For each hidden layer, the algorithm calculates:
- How much the error signal from the subsequent layer depends on the output of the current neuron.
- How much the error signal depends on the weights and biases of the current neuron.
- How much the error signal depends on the inputs to the current neuron (which were the outputs of the previous layer).
4. Accumulate Gradients: This process generates a gradient for every single weight and bias in the entire neural network, indicating the direction and magnitude of change needed to reduce the loss.
5. Update Weights and Biases (Optimization): Once all gradients are computed, an optimizer algorithm uses these gradients to update the weights and biases of the network. These updates are typically small steps in the direction opposite to the gradient (gradient descent), aiming to minimize the loss function.
The cycle of Forward Propagation and Backpropagation is repeated over many iterations (epochs) and mini-batches of data. With each cycle, the network's weights and biases are refined, gradually reducing the overall loss and improving the accuracy of its predictions.