Variance reduction in lattice QCD observables via normalizing flows
Authors: Ryan Abbott, Denis Boyda, Yang Fu, Daniel C. Hackett, Gurtej Kanwar, Fernando Romero-López, Phiala E. Shanahan, Julian M. Urban
Organizations: 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 · Fermi National Accelerator Laboratory, Batavia, IL 60510, U.S.A. · Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, EH9 3FD Edinburgh, United Kingdom · Albert Einstein Center, Institute for Theoretical Physics, University of Bern, 3012 Bern, Switzerland
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.
Figures & tables
Model
Theory
Parameters
Train geom.
Train ESS / E2
Eval. geom.
Eval. ESS / E2
A
SU(3) YM
β=6.0 , λ=2⋅10−3
43×32
99.996% / 10
163×32
99.6% / 950
B
Nf=2 QCD
β=5.6,κ=0.1577,λ=2⋅10−3
43×32
99.996% / 10
163×32
98.5% / 3800
B+PF
98.8% / 3000
Table 1: Details of the trained models used for the demonstration of glueball correlator computation. For Model B, the two rows of evaluation metrics correspond to evaluations with naive pseudofermions (“B”) and with a pseudofermion flow model (“B+PF”).
Figure 1: Illustration of variance reduction in the glueball correlator in the SU(3) Yang-Mills theory with β=6 for a 163×32 lattice geometry. The upper panel displays the correlator itself, evaluated on an ensemble of 3200 configurations. The baseline calculation using the standard two-point function evaluation is compared against the derivative-trick estimate evaluated using both the finite flow with λ=2⋅10−3 and the unbiased linearized estimate. In the lower panel, the ratio between the variance of the baseline and flow-improved estimates is shown.
Figure 2: Variance reduction in the glueball correlator achieved by the flow approach at timeslice t=2a as a function of the L3×32 lattice geometry. Results for both the finite- λ and linearized approach are shown. Data for the L=16 case is obtained from an ensemble of 3200 configurations while the remaining cases are evaluated using ensembles of approximately 1600 configurations each.
Figure 3: Illustration of the variance reduction in the glueball correlator in QCD with Nf=2 Wilson fermions using 1280 configurations. The upper panel displays the correlator itself; the flow-improved estimates using both naive pseudofermions and a pseudofermion flow are compared against a baseline calculation using the standard computation of the two-point function. A single pseudofermion hit for both flow estimates. In the lower panel, the ratio of variances is shown with and without employing a pseudofermion flow.
Model
Theory
Parameters
Train geom.
Train ESS / E2
Eval. geom.
Eval. ESS / E2
C
QCD
κ=0.164111,β=3.8,μ=0.012,λ=2×10−3
44
99.94% / 150
163×32
65.0% / 1.3×105
C+PF
69.2% / 1.1×105
Table 2: Details of the trained models used for the demonstration of hadronic matrix element calculation via the Feynman–Hellmann theorem. The two rows of evaluation metrics correspond to evaluations with naive pseudofermions (“C”) and with a pseudofermion flow model (“C+PF”).
Figure 4: Illustration of the variance reduction in the gluon momentum fraction in QCD with Nf=2 Wilson fermions. The upper panel compares the gluon momentum fraction of the pion computed using flows (orange squares) versus the baseline ϵ -reweighting method (blue circles), both evaluated on the same ensemble of 8576 configurations. In the lower panel, the ratio between the variance of the baseline and the flow-improved estimate is shown. The results are shown as a function of the lower end of the fit range used for the extraction of the mass, tmin .
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
Neural generative samplers for lattice field theory can be costly to train and evaluate. When they miss modes or assign them incorrect relative weights, biased observables do not reveal which collective variables are responsible. We project a trained flow-matching velocity onto vector fields built from lattice operators and Fourier modes. In two-dimensional lattice φ4 theory, the projection separates changes in the overall magnetization from the lowest nonzero-momentum fluctuations and guides an explicit invertible proposal that treats them separately. Allowing the amplitude of the lowest nonzero-momentum fluctuations to depend on the magnetization improves the overlap between the proposal and target distributions, while the same two-parameter modification at higher momenta gives smaller improvements. The Metropolis--Hastings correction defines a Markov chain with the target Boltzmann distribution as its stationary law, and the normalized proposal density yields finite-volume partition-function estimates consistent with an independent HMC calculation. At the larger tested volume, the overlap between the proposal and target distributions deteriorates substantially, limiting the range over which the same parameterization remains effective.
Moxian Qian
Johannes Gutenberg University Mainz, 55128 Mainz, Germany
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.
Michael S. Albergo, Gurtej Kanwar
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