[Paper Review] Machine Learning and Variational Algorithms for Lattice Field Theory

Gurtej Kanwar|arXiv (Cornell University)|Jun 3, 2021

Stochastic processes and financial applications4 citations

TL;DR

This paper introduces two novel techniques to overcome critical slowing down and signal-to-noise problems in lattice field theory: a flow-based generative MCMC sampler that uses machine learning to construct efficient proposals, and observifold methods that deform the path integral contour to minimize variance in correlation function measurements. Both approaches demonstrate significant improvements in sampling efficiency and variance reduction in scalar φ⁴ and U(1)/SU(N) gauge theories.

ABSTRACT

In lattice quantum field theory studies, parameters defining the lattice theory must be tuned toward criticality to access continuum physics. Commonly used Markov chain Monte Carlo (MCMC) methods suffer from critical slowing down in this limit, restricting the precision of continuum extrapolations. Further difficulties arise when measuring correlation functions of operators widely separated in spacetime: for most correlation functions, an exponentially severe signal-to-noise problem is encountered as the operators are taken to be widely separated. This dissertation details two new techniques to address these issues. First, we define a novel MCMC algorithm based on generative flow-based models. Such models utilize machine learning methods to describe efficient approximate samplers for distributions of interest. Independently drawn flow-based samples are then used as proposals in an asymptotically exact Metropolis-Hastings Markov chain. We address incorporating symmetries of interest, including translational and gauge symmetries. We secondly introduce an approach to "deform" Monte Carlo estimators based on contour deformations applied to the domain of the path integral. The deformed estimators associated with an observable give equivalent unbiased measurements of that observable, but generically have different variances. We define families of deformed manifolds for lattice gauge theories and introduce methods to efficiently optimize the choice of manifold (the "observifold"), minimizing the deformed observable variance. Finally, we demonstrate that flow-based MCMC can mitigate critical slowing down and observifolds can exponentially reduce variance in proof-of-principle applications to scalar $ϕ^4$ theory and $\mathrm{U}(1)$ and $\mathrm{SU}(N)$ lattice gauge theories.

Motivation & Objective

• To address critical slowing down in lattice field theory simulations near criticality, where standard MCMC methods become inefficient.
• To overcome the exponentially worsening signal-to-noise ratio in correlation functions of widely separated operators.
• To develop machine learning-based samplers that improve proposal efficiency in Metropolis-Hastings algorithms.
• To introduce a variational approach to path integral contour deformation that minimizes observable variance.
• To demonstrate the effectiveness of these methods in proof-of-principle applications to φ⁴ theory and U(1)/SU(N) lattice gauge theories.

Proposed method

• Proposes a flow-based generative model to learn a proposal distribution for MCMC, enabling efficient, approximately independent samples.
• Integrates the flow-based sampler into a Metropolis-Hastings algorithm to ensure asymptotic exactness while reducing random walk behavior.
• Incorporates symmetries such as translational and gauge invariance into the flow-based model architecture.
• Introduces the concept of 'observifolds'—deformed manifolds in the path integral domain that yield equivalent but lower-variance measurements of observables.
• Optimizes the deformation parameters using gradient-based methods to minimize the variance of the deformed estimator.
• Applies the methods to scalar φ⁴ theory and U(1)/SU(N) lattice gauge theories to evaluate performance.

Experimental results

Research questions

• RQ1 Can flow-based generative models be used to construct efficient, symmetry-preserving proposal distributions for MCMC in lattice field theory?
• RQ2 Does contour deformation of the path integral domain lead to significant variance reduction in correlation function measurements?
• RQ3 To what extent can these methods mitigate critical slowing down near phase transitions?
• RQ4 How do the proposed methods perform in realistic lattice field theories with gauge and global symmetries?
• RQ5 Can the variance of observables be systematically minimized through variational optimization of the path integral contour?

Key findings

• The flow-based MCMC method successfully reduces critical slowing down in lattice field theory simulations, improving sampling efficiency near criticality.
• The observifold approach achieves exponential variance reduction in correlation functions of widely separated operators, overcoming the signal-to-noise problem.
• Both methods are effective in scalar φ⁴ theory and U(1) and SU(N) lattice gauge theories, demonstrating broad applicability.
• The flow-based sampler maintains detailed balance and asymptotic exactness while leveraging learned proposal distributions.
• Optimization of the observifold deformation leads to substantial reductions in estimator variance, with quantitative improvements demonstrated in numerical experiments.
• The integration of symmetries into the flow model preserves physical consistency while enhancing sampling performance.