  We explain how to use diffusion models to learn inverse renormalization group
flows of statistical and quantum field theories. Diffusion models are a class
of machine learning models which have been used to generate samples from
complex distributions, such as the distribution of natural images. These models
achieve sample generation by learning the inverse process to a diffusion
process which adds noise to the data until the distribution of the data is pure
noise. Nonperturbative renormalization group schemes in physics can naturally
be written as diffusion processes in the space of fields. We combine these
observations in a concrete framework for building ML-based models for studying
field theories, in which the models learn the inverse process to an
explicitly-specified renormalization group scheme. We detail how these models
define a class of adaptive bridge (or parallel tempering) samplers for lattice
field theory. Because renormalization group schemes have a physical meaning, we
provide explicit prescriptions for how to compare results derived from models
associated to several different renormalization group schemes of interest. We
also explain how to use diffusion models in a variational method to find ground
states of quantum systems. We apply some of our methods to numerically find RG
flows of interacting statistical field theories. From the perspective of
machine learning, our work provides an interpretation of multiscale diffusion
models, and gives physically-inspired suggestions for diffusion models which
should have novel properties.
