[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