Diagram of a neural network showing input features flowing through hidden layers to output probabilities, paired with the six training steps: linear layer, activation, stacked layers, softmax and loss, backpropagation, optimizer update

Neural Networks From Scratch: Layer by Layer

A neural network is a chain of functions. Each layer takes a tensor in, does one simple mathematical operation, and passes a tensor out. There’s no magic in it. It’s linear algebra, a nonlinearity, and calculus, repeated. Here’s the chain in the order data actually flows through it: forward pass first, then how it learns. Every equation gets paired with its exact PyTorch equivalent, so you can see where the math actually lives in code. ...

July 1, 2026 · 11 min · 2160 words · Rahul Bhati