Flow matching

Flow matching is a generative AI approach that builds upon diffusion to denoise Gaussian noise into a complex distribution of interest. In contrast with diffusion, flow matching learns a velocity operation to push samples along a path from a (usually) normal distribution to an arbitrarily complex distribution.

Details

Flow matching uses learned velocity operations $v_{\theta }(x_t,t)$ to convert a set of normally distributed noise $p_0$ to a data distribution $p_D$. The generative process proceeds by stepwise integration of the ordinary differential equation: $d{x}_t=v_{\theta }(x_t,t)dt$.

During training, the data distribution consists of $x\sim p_D$ and noise samples $ϵ\sim N(0,\mathbf{I})$. Training data at time point $t$ is just a linear combination of ground truth and noise $x_t=tx+ϵ\ast (1-t)$. The network being trained tries to reproduce this velocity and movement. For example, SimpleFold uses L2 regression: ${\ell }_{\mathrm{FM}}={\mathbb{E}}_{x,s,ϵ,t}\left[\frac{1}{N_a}{\Vert v_{\theta }(x_t,s,t)-(x-ϵ)\Vert }^2\right]$.