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

2025-10-01

AI Summary

The sign problem—arising from complex-valued actions in quantum field theory and hindering Monte Carlo sampling via complex Langevin dynamics—remains a fundamental challenge. Method: This work introduces, for the first time, score-based and energy-based diffusion models into complex configuration space to implicitly learn and reconstruct the underlying non-equilibrium, non-real-valued probability distributions. Training data are generated via complex Langevin dynamics; both model classes are trained separately for distribution estimation and sample reconstruction. Theoretical analysis establishes their representational capacity for such complex, non-Hermitian distributions. Contribution/Results: Experiments demonstrate that the learned models accurately approximate the target complex Langevin distribution, achieving substantial improvements in sampling efficiency and numerical stability. This work establishes a novel machine-learning paradigm for tackling the sign problem in quantum field theory and extends the theoretical foundations and physical applicability of generative diffusion models to complex-probability systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Solving theories with complex action sign problems
• Understanding probability distributions from complex Langevin processes
• Comparing diffusion models for learning complex Langevin distributions

Innovation

• Combining complex Langevin dynamics with diffusion models
• Learning distributions sampled by complex Langevin process
• Comparing score-based and energy-based diffusion models