[Paper Review] Renormalizing Diffusion Models

Jordan Cotler, Semon Rezchikov|arXiv (Cornell University)|Aug 23, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper introduces a novel framework that uses diffusion models to learn inverse renormalization group (RG) flows in statistical and quantum field theories, treating RG schemes as diffusion processes. By training models to reverse these noise-adding processes, the method enables adaptive bridge sampling for lattice field theory and provides a variational approach to compute ground states of quantum systems with physically motivated, multiscale architectures.

ABSTRACT

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.

Motivation & Objective

• To bridge machine learning and quantum/statistical field theory by framing RG flows as diffusion processes.
• To develop a method for learning inverse RG flows using diffusion models, enabling efficient sampling in lattice field theories.
• To provide a variational framework for computing ground states of quantum field theories using score-based generative models.
• To offer physically motivated, multiscale diffusion model architectures that respect the structure of effective field theories.
• To enable cross-comparison of results across different RG schemes through a unified, interpretable framework.

Proposed method

• Formalizes nonperturbative RG schemes as diffusion processes in field space, with noise addition corresponding to coarse-graining.
• Uses score-based generative modeling to learn the reverse process—i.e., denoising—thereby reconstructing the original field distribution.
• Applies normalizing flows to variational inference, optimizing proposal distributions for efficient sampling in lattice field theories.
• Introduces a continuous-time formalism for RG flows, modeling them as Fokker-Planck-type equations with functional derivatives.
• Constructs multiscale diffusion models by embedding physical RG kernels (e.g., Polchinski, Carosso) into the noise schedule and score function.
• Uses the inverse diffusion process to generate samples from the original field distribution, enabling computation of observables via Monte Carlo.

Experimental results

Research questions

• RQ1 Can diffusion models be used to invert explicit, physically motivated RG flows in lattice field theories?
• RQ2 How can diffusion models be adapted to serve as adaptive bridge samplers in lattice field theory with improved mixing?
• RQ3 Can the inverse diffusion process be used to variational approximate the ground state wavefunction of a quantum field theory?
• RQ4 What are the physical and statistical properties of diffusion models trained on different RG schemes (e.g., Polchinski vs. Carosso)?
• RQ5 How do the learned score functions and noise schedules reflect the multiscale structure of effective field theories?

Key findings

• The inverse diffusion process successfully reconstructs the original field distribution from a noisy, coarse-grained state, enabling accurate sampling in lattice field theories.
• The method produces adaptive bridge samplers by learning the inverse flow of a specified RG scheme, improving sampling efficiency over standard Markov chain Monte Carlo.
• The framework allows for direct comparison of results across different RG schemes (e.g., Polchinski and Carosso) due to a unified, physically grounded formulation.
• Diffusion models trained on the Polchinski and Carosso RG schemes reproduce known fixed points and phase transitions in $ ho^4$ theory, validating the approach numerically.
• The learned score functions encode multiscale field structure, with noise schedules reflecting the momentum-space cutoffs of the underlying RG kernel.
• The variational method using diffusion models successfully computes ground state wavefunctions for quantum systems, particularly where the wavefunction is real and positive.