  We establish that Polchinski's equation for exact renormalization group flow
is equivalent to the optimal transport gradient flow of a field-theoretic
relative entropy. This provides a compelling information-theoretic formulation
of the exact renormalization group, expressed in the language of optimal
transport. A striking consequence is that a regularization of the relative
entropy is in fact an RG monotone. We compute this monotone in several
examples. Our results apply more broadly to other exact renormalization group
flow equations, including widely used specializations of Wegner-Morris flow.
Moreover, our optimal transport framework for RG allows us to reformulate RG
flow as a variational problem. This enables new numerical techniques and
establishes a systematic connection between neural network methods and RG flows
of conventional field theories.
