A graph neural network updates every node from its own features and an order-invariant aggregate of its neighbours' — the only form a layer can take when node indices carry no meaning. That single constraint produces message passing, message passing produces the GCN layer, the choice of aggregator sets a hard ceiling on which neighbourhoods a layer can tell apart, and repeated averaging drives every representation toward the same degree-dominated fixed point.

Nothing in a transformer specifies what any attention head should look at — the query and key matrices start as random noise and every pattern is a fit to the training task. Train a two-layer transformer from scratch on a repeated-sequence task and the canonical circuit appears: a previous-token head in layer 0 writing each position's predecessor into the residual stream, and an induction head in layer 1 using it to complete the copy. Knock out either one and accuracy collapses from 99.9% to chance.

Q-learning estimates the value of every action in every state directly from experience, with no model of the environment. These notes work through the tabular version end to end — state design, reward shaping, the temporal-difference update, and ε-greedy exploration — using the Snake agent from this site's game page as the running example, retrained from scratch in a live demo.

Self-attention computes each token's new representation as a weighted average of every token's value vector, with the weights chosen by content: a softmax over query–key dot products. The 1/√d scaling is literally a softmax temperature, causal masking is a renormalized softmax over the prefix, and multiple heads run the same lookup in parallel subspaces so different relations can coexist. From Bahdanau's RNN attention to sinusoidal positions, the full residual + LayerNorm + FFN block, and the inference tricks — KV cache, GQA, FlashAttention — that make the O(n²) core affordable.

An implicit neural representation stores a signal as a function: a small network maps continuous (x, y) coordinates to RGB, so the image lives in the weights instead of a pixel grid. Why a plain ReLU MLP can only learn a blurry version (spectral bias), how Fourier features and SIREN's sine activations fix it, and what the functional view buys — rendering one trained network at any resolution, including scales it never saw. With live in-browser training, and a final demo that reruns the whole comparison on any image dropped into the page — processed entirely locally, nothing uploaded.

Hinton's Forward-Forward algorithm trains a network without backpropagation: it replaces the forward-then-backward sweep with two forward passes — one on real data, one on fake — and lets every layer learn immediately from its own local 'goodness' objective. What goodness is, why layer normalization is the load-bearing trick, and a live in-browser training race between Forward-Forward and backprop.

A World Models agent splits into a large world model trained unsupervised and a tiny controller trained by evolution: V compresses each frame to a latent vector, M learns to predict the next latent given the action, and C maps that compressed state to an action — small enough to train entirely inside the model's own hallucinated rollouts, then transfer to reality.

The temperature parameter in an LLM divides the logits before the softmax. Raise it and the output distribution flattens, so its entropy rises — and that is not a loose analogy but the literal statistical-mechanics relationship: a softmax is a Boltzmann distribution with energy = −logit, and temperature trades expected energy against entropy, monotonically, from a deterministic argmax to the uniform distribution.

The Black–Scholes formula — N(d₁), N(d₂) — is easy to memorise and hard to reconstruct. Rebuilt from the payoff up: an option's price is the discounted, risk-neutral expectation of its payoff, and volatility is what gives it value. Drag the strike, widen the volatility, watch a Monte-Carlo average crawl onto the Black–Scholes price, and read the Greeks as slopes and curvature.

How many flips does it take to confidently tell a rigged coin from a fair one? Flip a coin and watch the evidence pile up, see a p-value light up, push the false-alarm and missed-detection curves apart, then read the exact number off one clean formula — and confirm it with a thousand simulated experiments. The punchline: near-fair coins are expensive to expose, and the cost scales like 1/(p−½)².

Entropy is usually introduced as a formula — −Σ p log p — with the meaning left implicit. Rebuilt from the ground up: entropy is average surprise, measured in yes/no questions. Drag a coin's bias, reshape a distribution, and watch a sampler's running surprise converge onto the entropy.

Hopfield networks slide downhill into the nearest memory — and get stuck. Add a single ingredient, temperature, and that deterministic descent becomes a Boltzmann machine that samples, escapes local minima, and learns. A hands-on, visual walk-through with live energy landscapes, annealing, and a trainable RBM.

A Markov Decision Process models sequential decisions under uncertainty, where today's choice shapes tomorrow's options. An interactive introduction to MDPs and value iteration, worked through one small example: should a 9-year-old study, play, or rest after school?