Stochastic differential equation

Then we'll go back to kind of like interpreting diffusion models as normalizing flows.

This is the sort of idea of converting an SDE to an ODE that we briefly talked about, but we didn't have time to go into a lot of detail, which will allow us to compute likelihoods exactly because it's a flow model.

And then we'll talk about how to make sampling efficient.

So take advantage of the fact that once you view generation as solving some kind of ordinary differential equation or stochastic differential equation, then you can use advanced numerical methods to accelerate sampling.

What a score-based model would do instead is it would kind of like attempt to kind of like correct, because there are numerical errors, you're going to make some mistakes.

And so what a score-based model would do is it would try to basically fix the mistakes by running Langevin's for that time step.

So you can kind of like combine just regular sampling from a diffusion model where you would take a thousand different steps or even less with MCMC style sampling to kind of like correct the mistakes of a basic sort of like numerical SDE solver.

It's not normally distributed.

If you don't include it, your equations will drift completely detached from reality.

And with Ito's formula, the text finally unleashes the power to solve stochastic differential equations, or SDEs.

We are no longer just observing randomness.

We are building dynamic engines with it.

This is what runs the modern world.

When a quant on Wall Street prices a complex options derivative, they are using geometric Brownian motion, an SDE solved by Ito calculus.

When physicists model particle diffusion in a reactor, they use SDEs.

The text walks through models like the Bessel process, showing how randomness can be baked directly into the differential geometry of the universe.