Renormalization Group Flows: from Optimal Transport to Diffusion Models

          @misc{ pirsa_PIRSA:25040095,
            doi = {10.48660/25040095},
            url = {https://pirsa.org/25040095},
            author = {Cotler, Jordan},
            keywords = {},
            language = {en},
            title = {Renormalization Group Flows: from Optimal Transport to Diffusion Models},
            publisher = {Perimeter Institute},
            year = {2025},
            month = {apr},
            note = {PIRSA:25040095 see, \url{https://pirsa.org}}
          }
          

Abstract

We show that Polchinski’s equation for exact renormalization group flow is equivalent to the optimal transport gradient flow of a field-theoretic relative entropy. This gives a surprising information-theoretic formulation of the exact renormalization group, expressed in the language of optimal transport. We will provide reviews of both the exact renormalization group, as well as the theory of optimal transportation. Our techniques generalize to other RG flow equations beyond Polchinski's. Moreover, we establish a connection between this more general class of RG flows and stochastic Langevin PDEs, enabling us to construct ML-based adaptive bridge samplers for lattice field theories. Finally, we will discuss forthcoming work on related methods to variationally approximate ground states of quantum field theories.