[Verse 1] In the neural networks dancing through your mind Energy functions govern what we find E of x and theta, that's the guiding force Gradient flow descends along its course X dot equals negative del sub x of E This is how the system finds its harmony Rolling down the hills of energy space Until it settles in its resting place [Chorus] Symmetric weights give energy's embrace Asymmetric breaks the mathematical base Hopfield stores the patterns that we need Boltzmann's stochastic, planted like a seed Energy landscapes guide the neural flow Cortical columns, this is how they grow [Verse 2] Hopfield networks need their weights to match Wij equals Wji, that's the perfect catch Without this balance, energy won't hold The system spirals, chaos takes control Symmetric connections make the function real Gradients point the way that neurons feel Memory patterns carved in neural stone Each basin holds what the network's known [Chorus] Symmetric weights give energy's embrace Asymmetric breaks the mathematical base Hopfield stores the patterns that we need Boltzmann's stochastic, planted like a seed Energy landscapes guide the neural flow Cortical columns, this is how they grow [Verse 3] Boltzmann machines add the thermal noise Temperature gives neurons random choice Stochastic relaxation finds the way Through probability's intricate display Sigmoid activation, exponential might Negative energy determines what's right Thermal equilibrium, the final state Where all the probabilities relate [Bridge] Free energy principle rules the brain Variational inference breaks the chain Between what is and what we think we see Minimizing surprise, that's the key The cortex builds its models of the world Through energy functions, mysteries unfurled [Verse 4] Ising spins align in magnetic fields Binary states and what the coupling yields Phase transitions mark the critical point Where order emerges, systems conjoint Mean field theory approximates the dance Of many bodies in their thermal trance Magnetization curves through temperature's range Energy landscapes constantly rearrange [Chorus] Symmetric weights give energy's embrace Asymmetric breaks the mathematical base Hopfield stores the patterns that we need Boltzmann's stochastic, planted like a seed Energy landscapes guide the neural flow Cortical columns, this is how they grow [Outro] From gradient descent to thermal dance Mathematics gives the brain its chance To organize the chaos into thought Energy functions, this is what we've taught
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