Artificial Neural Network (ANN):

Computational model inspired by biological neurons. Network of interconnected units (neurons) that processes inputs via weighted sums and non-linear activations to learn mappings.

• Neuron: $net=\sum w_i x_i + b$, $y=f(net)$ where $f$ is activation (sigmoid $1/(1+e^{-net})$, tanh, ReLU $max(0,net)$).
• Architecture: Input layer → one or more hidden layers → output layer. Fully connected. Depth determines representation power.

Feedforward Algorithm:
Information flows forward only, no cycles.

1. Initialize weights small random.
2. For each layer $l$: $a^{(l)}=f(W^{(l)}a^{(l-1)}+b^{(l)})$
3. At output, compute loss $E=\frac12\sum(y-a^{(L)})^2$ or cross-entropy.
4. Example: 2-input XOR with hidden 2 neurons: $h_1=sigmoid(w_{11}x_1+w_{12}x_2+b_1)$, $output=sigmoid(v_1h_1+v_2h_2+c)$.

Backpropagation Algorithm:
Efficient gradient computation via chain rule to train weights.

1. Forward pass as above, store $net, a$.
2. Output error $\delta^{(L)}=(y-a^{(L)})f'(net^{(L)})$
3. Propagate backward: $\delta^{(l)}= (W^{(l+1)T}\delta^{(l+1)})\odot f'(net^{(l)})$
4. Gradients $\partial E/\partial W^{(l)}=\delta^{(l)}a^{(l-1)T}$
5. Update $W:=W+\eta\delta a^T$, $b:=b+\eta\delta$ (SGD).
6. Iterate over epochs until convergence.

Example with numbers: Single neuron with sigmoid, input 0.5, target 1, $w=0.3$, $net=0.15$, $o=0.537$, $E=0.107$, $\delta=(0.463)*0.248=0.115$, $\Delta w=0.1*0.115*0.5=0.0057$ → weight increases.

Feedforward computes prediction; backprop assigns blame and updates weights to reduce loss, enabling learning of non-linear boundaries.