Learning the Integral of a Diffusion Model (opens in new tab)

Diffusion and flow matching models generate samples by iterative denoising. Iterative denoising means passing input to the neural network, running a forward pass, and taking the output back as input and rerunning the neural network. Often you do this 100 times, which is slow and expensive.

Flow maps / consistency models / shortcut models instead try to learn to compress this iterative work into 1 forward pass. This makes inference 100x faster as you'd only need to run the neural net forward pass once. Beyond speeding up inference, there are other advanced benefits to this, such as improved ability to perform inference-time steering.

Mathematically, learning a flow map corresponds to learning to solve an ordinary differential equation, i.e., learning the time integral of the velocity field. This mathematical foundation provides the basis for various training objectives for learning flow maps, which involve self-referential identities or identities such as the transport equation, which are discussed in the blog post.

Hope that helps! I'm an ML researcher currently researching flow maps.

This provides a high-level overview of diffusion models, you know, the models behind Stable Diffusion, Gemini banana, etc.

I haven't read it carefully but I think it's pretty comprehensive. From SDE to Flow matching formulation, and different perspective of constructing the flow maps, i.e. x-formulation or v-formulation. It also deals with distillation and consistency, which is used to fast sampling.

Overall, it's a good read if you are new to the field.

Why not put it into an AI yourself? :) I'd rather we avoided a precedent of asking for it and N people replying with their own favorite AI version. The comments section would end up a ghost town.

Extreme TL;DR: Diffusion models are like getting f(x) by calculating and summing f'(0), f'(1)...f'(x). Flow models are like just calculating f(x).

MIT's OCW is usually pretty reliable:

https://www.practical-diffusion.org/lectures/

There is more math-heavy https://diffusion.csail.mit.edu/2026/index.html

I really liked Calvin Lou’s tutorial. It’s quite old at this point and probably not up to date, but I felt it was awesome for understanding the concepts

As explanation, something I wrote previously:

The most common approach to modeling continuous distributions is to train a reversible model f that maps it to another continuous distribution P that is already known. The original image can be recovered by tracking the bits needed to encode its latent, as well as the reverse path:

  −log P(f(x)) − log|det ∂f/∂x (x)|

This technique is known as normalizing flows, as usually a normal distribution is chosen for the known distribution. The second term can be a little hard to compute, so diffusion models approximate it by using a stochastic PDE for the mapping. When f is a solution to an ordinary differential equation,

  dx/dt = g(x)

then

  log|det ∂f/∂x (x)| = ∫ Tr(∂g(x)/∂x) dt = ∫ E_{ε∼N(0,I)} [εᵀ ∂g(x)/∂x ε] dt

The last equality is known as Hutchison's estimator. Switching to a stochastic PDE

  dx′ = g(x′)dt + ε(t)dW

and tracking the difference δx = x′ − x, the mean-squared error approximately satisfies

  d(δxᵀδx)/dt = 2δxᵀ ∂g(x)/∂x δx,

which is close to Hutchinson's estimator, but weighted a little strange.

I briefly covered that connection in an earlier blog post: https://sander.ai/2023/07/20/perspectives.html#flow ... but it's definitely something that might deserve a longer-form treatment at some point :)

Thanks for pointing this out! I'm not sure why, the files are still on my Dropbox, they must have changed the link format at some point? I've gone ahead and fixed them.