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Diffusion models learn distributions generated by complex Langevin dynamics

Diaa E. Habibi, Gert Aarts, Lingxiao Wang, Kai Zhou

TL;DR

The paper tackles the challenge of the sign problem by leveraging diffusion models to learn the distribution sampled by Complex Langevin dynamics on the complexified space. By training a score-based diffusion model on data produced by CL, the authors estimate the log-derivative $\nabla \log P(x,y)$ of the CL-distribution and use it to generate new configurations without solving the intractable Fokker–Planck equation. In two one-dimensional test cases—the Gaussian model with a complex mass $\sigma_0$ and a quartic model confined to a strip—the diffusion model accurately reproduces the learned score and key statistical properties (moments and cumulants), validating the approach. The method provides a new diagnostic and data-generation tool for analyzing CL distributions and could be extended to more complex lattice field theories, although it does not solve the sign problem itself.

Abstract

The probability distribution effectively sampled by a complex Langevin process for theories with a sign problem is not known a priori and notoriously hard to understand. Diffusion models, a class of generative AI, can learn distributions from data. In this contribution, we explore the ability of diffusion models to learn the distributions created by a complex Langevin process.

Diffusion models learn distributions generated by complex Langevin dynamics

TL;DR

The paper tackles the challenge of the sign problem by leveraging diffusion models to learn the distribution sampled by Complex Langevin dynamics on the complexified space. By training a score-based diffusion model on data produced by CL, the authors estimate the log-derivative of the CL-distribution and use it to generate new configurations without solving the intractable Fokker–Planck equation. In two one-dimensional test cases—the Gaussian model with a complex mass and a quartic model confined to a strip—the diffusion model accurately reproduces the learned score and key statistical properties (moments and cumulants), validating the approach. The method provides a new diagnostic and data-generation tool for analyzing CL distributions and could be extended to more complex lattice field theories, although it does not solve the sign problem itself.

Abstract

The probability distribution effectively sampled by a complex Langevin process for theories with a sign problem is not known a priori and notoriously hard to understand. Diffusion models, a class of generative AI, can learn distributions from data. In this contribution, we explore the ability of diffusion models to learn the distributions created by a complex Langevin process.

Paper Structure

This paper contains 7 sections, 21 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Complex Langevin dynamics
  3. Diffusion models
  4. Application to complex Langevin dynamics
  5. Gaussian model
  6. Quartic model
  7. Outlook

Figures (3)

  • Figure 1: Gaussian model with complex mass parameter $\sigma_0 = 1+i$: vector field $\nabla \log P(x,y)$ and lines of constant $P(x,y)$ as given by the analytical results (\ref{['eq:FPE-G']}, \ref{['eq:dlogPexact']}) (left) and as learnt by the diffusion model (right).
  • Figure 2: Quartic model with parameters $\sigma_0=1+i, \lambda=1$: solution $P(x,y)$ of the FPE obtained by a double expansion in Hermite functions Aarts:2013uza (left) and as learnt by the diffusion model (right).
  • Figure 4: Drift in complex Langevin dynamics (left) and the score as learnt by the diffusion model, including contour lines (right). These vector fields are different, as they should be. Parameters as above.