Combining complex Langevin dynamics with score-based and energy-based diffusion models

TL;DR

This work tackles the sign problem by using diffusion-models to learn the distribution sampled by complex Langevin dynamics on the complexified configuration space. It compares score-based and energy-based diffusion approaches, highlighting that SBMs yield non-conservative scores whereas EBMs provide a conservative energy from which the target distribution can be reconstructed and sampled, including via MCMC. Applied to a complex-valued quartic model, both approaches reproduce CL observables and, in the case of EBMs, enable direct sampling from the learned distribution, offering a promising route to study theories with sign problems and potentially extend to field theories. The results demonstrate that diffusion-based learning can provide new insights into the complexified sampling distributions that arise in CL, with practical implications for exploring sign-problem-laden theories.

Abstract

Theories with a sign problem due to a complex action or Boltzmann weight can sometimes be numerically solved using a stochastic process in the complexified configuration space. However, the probability distribution effectively sampled by this complex Langevin process is not known a priori and notoriously hard to understand. In generative AI, diffusion models can learn distributions, or their log derivatives, from data. We explore the ability of diffusion models to learn the distributions sampled by a complex Langevin process, comparing score-based and energy-based diffusion models, and speculate about possible applications.

Figures (9)

• Figure 1: Complex-valued quartic model with parameters $\sigma_0=1+i$ and $\lambda=1$: CL drift in the complex plane (left) and histogram $P(x,y)$ obtained by sampling the CL process (right) Aarts:2013uza.
• Figure 2: Learned scores in the quartic model at the end of the backward process, using a score-based (left) and energy-based (right) diffusion model.
• Figure 3: Quartic model using the score-based formulation: Curl of the score, averaged over 10 independently trained models (left) and histogram obtained by sampling data using the process learnt by the score-based model (right).
• Figure 4: Quartic model: Energy $E_\theta({\mathbf x})$ learned in the energy-based diffusion model (left) and the corresponding distribution $p_\theta({\mathbf x})\sim \exp[-E_\theta({\mathbf x})]$ (right).
• Figure 5: Quartic model: Contour plot of the energy learned in the energy-based model.