A Monte Carlo estimator of flow fields for sampling and noise problems
Authors: Michael S. Albergo, Gurtej Kanwar
Organizations: Society of Fellows, Harvard University 78 Mount Auburn St, Cambridge, MA United States · Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, EH9 3FD Edinburgh, United Kingdom
Learned field transformations may help address ubiquitous critical slowing down and signal-to-noise problems in lattice field theory. In the context of an annealed sequence of distributions, field transformations are defined by integrating flow fields that exactly solve a local transport problem. These proceedings discuss a new Monte Carlo approach to evaluating these flow fields, which can then be used directly in such contexts or as a means of generating unbiased training data for machine learning approaches. By defining the Monte Carlo estimator using coupled Langevin noise, the statistical noise in the required integrals is significantly mitigated. Demonstrations of the method include a U(1) transport problem and an SU(N) glueball correlator.
Figures & tables
Figure 1: Evaluation of φt(ϕ=0.5) for the one-variable Gaussian distribution at t=0 described in the main text. Results are shown for 1024 Langevin walkers Φτ initialized from Φ0=0.5 . The right plot compares the cumulative integral as a function of the maximum integration limit T against the true value, demonstrating both convergence and the growing statistical noise expected from integrating the noisy integrand.
Figure 3: Left: Illustration of Langevin evolution with coupled noise in a quartic potential, showing eventual convergence of various initial conditions. Right: The dependence on the initial conditions, ∂Φτ/∂ϕ , which decreases exponentially when measured for the same trajectories.
Figure 4: Estimate of b(θ) using the Feynman-Kac method for the von Mises distribution defined in the main text. Each point in the left panel is evaluated using 4096 Langevin walkers and is plotted for a range of upper integration limits T . The right panels show detailed convergence as a function of T for two values of θ . Evolution of block-averaged sample estimates are shown by light gray traces, and the bootstrap average over all walkers is indicated by the dark gray error bars.
Figure 5: Evaluation of a 0++ glueball correlator as a function of separation x/a for an SU(2) lattice gauge theory as described in the main text. Estimates using a standard vacuum-subtracted estimator are compared to the flow-field approach with a Feynman-Kac estimator integrated up to a variety of integration limits T .
The generating functional in quantum field theory provides the natural framework for constructing correlation functions as derivatives with respect to source operators. We present a methodology that leverages machine-learned normalizing flows to reduce the variance of arbitrary N-point correlation functions of bosonic operators in lattice gauge field theory calculations by encoding a representation of the generating functional. We show that it is possible to systematically approach noiseless estimators of correlation functions in this framework. We demonstrate this methodology with applications to calculations of glueball correlation functions and Wilson loops in Quantum Chromodynamics and Yang-Mills theory. The results show up to three orders of magnitude variance reduction.
Ryan Abbott, Yang Fu, Daniel C. Hackett +3
1Physics Department, Columbia University, New York, NY 10027, USA · Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · 3The NSF AI Institute for Artificial Intelligence and Fundamental Interactions +3
We present a first study of a diffusion-based approach to accelerated sampling of the Nf=2 lattice Schwinger model. Our work is inspired by recent and growing successes in developing such generative models for ensemble generation in LFT to overcome the well-known critical slowing down problem. We train a U(1)-equivariant score-based generative model to sample gauge link configurations from the marginal Schwinger model. By computing model likelihoods, we obtain unbiased estimates for observables that closely match those produced by MCMC simulations. We also demonstrate improvement over HMC as measured qualitatively by a reduction in topological freezing near critical parameters.
Octavio Vega, Aida X. El-Khadra
Department of Physics University of Illinois Urbana-Champaign Urbana, IL 61801
Normalizing flows can be used to construct unbiased, reduced-variance estimators for lattice field theory observables that are defined by a derivative with respect to action parameters. This work implements the approach for observables involving gluonic operator insertions in the SU(3) Yang-Mills theory and two-flavor Quantum Chromodynamics (QCD) in four space-time dimensions. Variance reduction by factors of 10-60 is achieved in glueball correlation functions and in gluonic matrix elements related to hadron structure, with demonstrated computational advantages. The observed variance reduction is found to be approximately independent of the lattice volume, so that volume transfer can be utilized to minimize training costs.
Ryan Abbott, Denis Boyda, Yang Fu +5
Physics Department, Columbia University, New York, NY 10027, USA · Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · The NSF AI Institute for Artificial Intelligence and Fundamental Interactions +3