teach me all about neural networks and also code one from scratch

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Neural networks are computational models inspired by the human brain. A biological neuron receives signals through dendrites, processes them in the cell body, and sends output through the axon. Similarly, an artificial neuron takes multiple inputs, multiplies them by weights, adds a bias term, and applies an activation function to produce an output. The mathematical formula shows this as y equals f of the sum of weighted inputs plus bias. Neural networks are built by connecting multiple layers of neurons. The input layer receives raw data, hidden layers process and transform the information, and the output layer produces the final results. Each layer performs matrix operations, multiplying inputs by weights and adding biases. The network depth refers to the number of layers, while width refers to neurons per layer. Information flows forward through fully connected layers, where each neuron connects to all neurons in the next layer. Activation functions are crucial components that introduce non-linearity to neural networks. The sigmoid function maps inputs to values between 0 and 1, making it useful for probability outputs. ReLU, or Rectified Linear Unit, outputs zero for negative inputs and the input value for positive inputs, making it computationally efficient. The hyperbolic tangent function maps inputs to values between negative 1 and 1, providing zero-centered outputs. Without non-linear activation functions, neural networks would only be able to learn linear relationships, severely limiting their capability to model complex patterns. Neural network training is an iterative optimization process. It begins with a forward pass where input data flows through the network to generate predictions. Next, a loss function measures the difference between predictions and actual targets. Common loss functions include mean squared error for regression and cross-entropy for classification. The backward pass then computes gradients using backpropagation, determining how much each weight contributed to the error. Finally, gradient descent updates the weights by moving them in the direction that reduces the loss. This cycle repeats for many epochs until the network converges to optimal parameters. Backpropagation is the mathematical foundation of neural network training. It uses the chain rule of calculus to efficiently compute gradients of the loss function with respect to each weight. The process starts by computing the gradient of the loss with respect to the output, then propagates this error backward through each layer. For each connection, we multiply the incoming gradient by the local derivative of the activation function and the input value. This gives us the gradient for each weight, which we use to update parameters via gradient descent. The chain rule allows us to decompose complex derivatives into simpler components, making the computation tractable even for deep networks.