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Ergodic topology sampling toward the continuum with Stochastic Normalizing Flows
Claudio Bonanno, Andrea Bulgarelli, Elia Cellini, Alessandro Nada, Dario Panfalone, Davide Vadacchino, Lorenzo Verzichelli
TL;DR
Topological freezing at fine lattice spacings threatens ergodic sampling of topology in SU(3) Yang–Mills theory. This work studies a strategy combining open boundaries with Stochastic Normalizing Flows. Its scaling results indicate that the approach mitigates sampling costs, with Q2 autocorrelation times following a power law with a much smaller exponent than standard MCMC.
Problem
At fine lattice spacing, standard MCMC can become trapped in a topological sector, threatening ergodicity and reliable calculation of topological observables.
Source: §1 · pp. 1–2Method
Results
The flow-based benchmark follows a power law with exponent about 3.6, versus standard MCMC behavior compatible with an exponential law or a power law with exponent 7.
Source: A. Scaling of costs in the continuum · p. 8Takeaways & Limitations
The results indicate that the SNF-based approach effectively mitigates the scaling of costs for sampling topological observables toward the continuum limit.
Source: A. Scaling of costs in the continuum · p. 8
Abstract
from arXiv · showhide
In this work we provide numerical evidence that topological freezing in $\mathrm{SU}(3)$ Yang-Mills theory can be mitigated with a flow-based approach, as tested at lattice spacings as fine as 0.03~fm. We do so by analyzing the scaling of the costs of a computational strategy based on the combination of open boundary conditions and Stochastic Normalizing Flows. We show that the integrated autocorrelation time of the squared topological charge and the statistical uncertainty on the topological susceptibility remain under control throughout the explored parameter range with currently available computational resources. This marks an important step toward the use of even more efficient flow-based architectures for ergodic sampling of topology in state-of-the-art lattice gauge theory calculations.
1. INTRODUCTION
As the continuum limit is approached, topological sectors become harder for standard MCMC to cross, threatening ergodicity and reliable estimates of topological observables. This study examines whether Stochastic Normalizing Flows can mitigate that freezing in realistic four-dimensional gauge-theory calculations.
1. INTRODUCTION
- At fine lattice spacing, growing barriers between topological sectors can trap standard MCMC in one sector, causing topological freezing and loss of ergodicity.This can impair topological observables and potentially bias other expectation values through power-law-suppressed finite-size effects.Source: §1 · pp. 1–2
- The integrated autocorrelation time of powers of topological charge, often Q2, tracks the severity of freezing as lattice spacing shrinks.Its growth may be a power law with a larger dynamical critical exponent than local observables, or exponential [17] [18] [20] [21] [22].Source: §1 · p. 1
- Open-boundary approaches and non-equilibrium Monte Carlo are among the strategies explored to mitigate topological freezing.Open boundaries can improve topology autocorrelations, while boundary tempering transfers improved scaling to periodic systems [54] [55] [56] [57].Source: §1 · p. 2
- Normalizing Flows can connect a tractable prior to a target distribution, but their training costs make them not yet viable for state-of-the-art four-dimensional gauge-theory sampling.Appropriate reweighting or accept-reject procedures ensure asymptotic exactness.Source: §1 · p. 2
- Stochastic Normalizing Flows alternate stochastic non-equilibrium Monte Carlo updates with learned transformations, retaining favorable scaling and potentially reducing sampling costs and training costs relative to NEMC.The paper presents a scaling study in realistic conditions and reports that Q2 autocorrelation times and uncertainties on ⟨Q2⟩ remain under control across the explored range.Source: §1 · p. 2
2. THEORETICAL AND NUMERICAL SETUP
The study uses pure SU(3) Yang–Mills theory with a boundary defect whose coupling is varied to interpolate between open and periodic boundaries. It samples a prior with standard local updates and uses Wilson cooling to measure topological charge on smoothed fields.
2. THEORETICAL AND NUMERICAL SETUP
- The boundary-defect coupling λn varies from 0 for open boundaries to 1 for periodic boundaries over nstep out-of-equilibrium steps.The defect consists of temporal gauge links on a cubic boundary region; each trajectory occurs every nbetween standard updates of the open-boundary system.Source: §2 · p. 3
- The prior uses a 1:4 combination of Cabibbo–Marinari heat-bath and over-relaxation updates.The paper calls this combination the standard algorithm.Source: §2 · p. 3
- Topological charge is measured with the clover discretization on smoothed fields to avoid the discretization’s multiplicative and additive renormalizations.Wilson cooling is chosen for its extremely low computational cost; its smoothing radius is related to gradient-flow time by Rs = a√(8t).Source: §2 · p. 3
3. A FLOW-BASED APPROACH TO TOPOLOGICAL FREEZING
The approach combines an open-boundary prior, used to decorrelate topological charge, with a flow that removes unwanted finite-size effects from the defect to provide exact, inexpensive sampling with periodic boundaries.
3. A FLOW-BASED APPROACH TO TOPOLOGICAL FREEZING
- The strategy uses an open-boundary prior to decorrelate topological charge, then applies an SNF flow to remove defect-induced finite-size effects for exact, cheap periodic-boundary sampling.The flow-based approach is presented as the means of obtaining exact and inexpensive sampling with periodic boundaries.Source: §3 · p. 3
A. Stochastic Normalizing Flows for boundary conditions
SNFs address dissipation in transforming open-boundary configurations to periodic-boundary configurations by interleaving stochastic updates with trainable invertible transformations. In this construction, defect-localized coupling layers transport configurations between intermediate distributions while reducing protocol dissipation.
A. Stochastic Normalizing Flows for boundary conditions
- NEMC transforms open-boundary configurations to the periodic target exactly, but fast evolution can yield broadly distributed reweighting factors and nearly reversible evolution may require many Monte Carlo steps.SNFs interleave stochastic updates with deterministic, invertible, trainable transformations to improve this construction.Source: A. Stochastic Normalizing Flows for… · pp. 3–4
- Each SNF step combines a detailed-balance Monte Carlo transition for the intermediate action with an invertible trainable coupling layer.The stochastic updates anneal between boundary conditions; deterministic layers transport configurations toward the next intermediate distribution and reduce protocol dissipation.Source: A. Stochastic Normalizing Flows for… · p. 4
- Defect coupling layers transform only links on or near the defect, leaving links outside that neighborhood unchanged and keeping computational cost much smaller than for a global transformation.The construction uses eight even–odd and directional masks, preserves gauge equivariance, and yields a tractable Jacobian.Source: A. Stochastic Normalizing Flows for… · p. 4
B. Training and sampling
SNFs combine stochastic updates with invertible transformations to move configurations between distributions; training minimizes dissipated work to improve reweighting efficiency. Transferred coupling layers retain an approximately threefold efficiency gain over standard NEMC, while training costs are negligible compared with sampling.
B. Training and sampling
- The generalized work supplies the reweighting factor for unbiased PBC expectation values from forward OBC-to-PBC SNF evolutions.The observable is calculated at the last step of the evolution.Source: B. Training and sampling · p. 5
- The ESS estimator ranges from 0 for infinitely noisy reweighting to 1 for a perfectly efficient estimator.ESS is defined as the theoretical variance of an observable divided by the variance of its estimator.Source: B. Training and sampling · p. 5
- Training minimizes average generalized work because the free-energy difference is independent of the coupling-layer parameters, reducing weight fluctuations and increasing ESS.Each of the nstep terms is minimized independently, leaving the protocol to dictate the path while each step's efficiency is optimized.Source: B. Training and sampling · p. 5
- Approximately 3× the efficiency of standard NEMC is retained when trained coupling layers are transferred to large lattices.The result supports the view that defect coupling layers act locally and are unaffected by global lattice properties at this expressiveness.Source: B. Training and sampling · p. 5
- Training uses about 1000–2000 gradient-descent steps per β and Ld/a, and its costs are negligible relative to sampling.The authors caution that larger loops or deeper smearing may transfer less effectively to larger volumes and be less amenable to interpolation.Source: B. Training and sampling · p. 5
A. Scaling of costs in the continuum
The SNF benchmark cost grows with a much smaller power-law exponent than standard MCMC in the continuum scaling study. The comparison suggests mitigation of topological-observable sampling costs, though the finest-lattice autocorrelation estimate is less precise.
A. Scaling of costs in the continuum
- Integrated autocorrelation times for Q2 remain remarkably constant under the chosen scaling strategy, although the finest-lattice estimate is less accurate because of limited samples.The authors leave a possible dependence on nstep and additional decorrelation during mixed-boundary evolution for future study.Source: A. Scaling of costs in the continuum · p. 6, Figure 2 · p. 6
- Increasing nstep as the lattice spacing decreases keeps average dissipated work and ESS remarkably constant, with only a weak rise in dissipation and fall in ESS.The weak trends are compatible with the defect coupling λnβ needing larger values toward the continuum limit.Source: A. Scaling of costs in the continuum · p. 7, Figure 4 · p. 7
- The SNF benchmark cost follows a power law with exponent about 3.6, while standard MCMC is compatible with exponential growth or a power law with a very large exponent.The authors describe this as clear evidence that their SNF approach mitigates continuum-limit costs, while noting that more rigorous study is needed for fine details.Source: Figure 5 · p. 7, A. Scaling of costs in the continuum · p. 8
- SNFs are more expensive down to a/√8t0 ∼0.1 (a ∼0.048fm), but may become more efficient below a/√8t0 ∼0.08 (a ∼0.038fm).Standard-MCMC τint(Q2) is difficult to determine reliably with limited statistics, including a very large error at β = 6.6.Source: A. Scaling of costs in the continuum · p. 8
- The main computational cost of the SNF approach comes from nstep, which sets the coefficient of its benchmark scaling law.The authors suggest that reducing dissipation through improved defect coupling layers could lower the scaling curve while preserving its scaling with a.Source: A. Scaling of costs in the continuum · p. 8
B. Topological susceptibility at fine spacings
Fine-lattice SNF determinations of the topological susceptibility have roughly 4.5% relative errors for β = 6.40–6.70 and agree within larger error bars with the coarser-lattice continuum extrapolation. The comparison uses a fixed smoothing radius and tests control of topology-related uncertainties rather than a new precision determination of χ.
B. Topological susceptibility at fine spacings
- At β = 6.40, 6.50, 6.60, and 6.70, χ has ∼4.5% relative error; at β = 6.80, uncertainty is ∼9%.The β = 6.80 ensemble has about 4k samples, versus about 10k for the others, and larger residual autocorrelations.Source: B. Topological susceptibility at fine… · p. 8
- The study targets control of topology autocorrelations and susceptibility uncertainties, not a novel precision determination of χ.The coarser-lattice reference data have about 0.3%–0.6% precision.Source: B. Topological susceptibility at fine… · p. 8
- All data use a fixed cooling smoothing radius Rs/√(8t0) ≃ 0.918, corresponding to gradient-flow time t/t0 ≃ 0.843.The matched-radius data were obtained by cubic-spline interpolation of a^4χ versus ncool.Source: B. Topological susceptibility at fine… · p. 8
- Fine-spacing SNF values for χ agree within their larger error bars with the coarser-spacing continuum extrapolation, deviating by at most 2.5 standard deviations.The extrapolation uses O(a^2) leading lattice artifacts; the comparison is with high-statistics standard-algorithm results.Source: B. Topological susceptibility at fine… · p. 8, Figure 6 · p. 8
5. CONCLUSIONS
The study compares the real cost scaling of open-boundary and Stochastic Normalizing Flow sampling with the MCMC baseline, finding milder cost growth toward the continuum. The approach remains expensive in absolute terms, while combinations with PTBC, multiscale samplers, and other physical settings are proposed as future directions.
5. CONCLUSIONS
- Combining open boundaries with Stochastic Normalizing Flows yields milder growth of topological-observable costs toward the continuum than standard MCMC.Residual autocorrelation times scale approximately as a^-2, while SNF removal of boundary effects costs approximately a^-3.Source: §5 · pp. 8–9
- The strategy remains expensive in absolute terms up to relatively fine lattice spacings because its favorable scaling has large coefficients.The proposed improvement directions target dissipation through boundary-condition parameter choices and more expressive defect coupling layers.Source: §5 · p. 9
- Combining NEMC/SNFs with PTBC is proposed as a future direction; flow-based methods could reduce the number of concurrently simulated replicas while preserving favorable continuum scaling.A quantitative cost comparison with PTBC at very fine spacings is beyond this work’s scope and complicated by different memory requirements.Source: §5 · p. 9
- Combining this topology-focused approach with hierarchical multiscale samplers could address both topological freezing and critical slowing associated with long-distance modes and increasing volumes.The paper presents this combination as a promising strategy, not an evaluated result.Source: §5 · p. 9
- The NEMC/SNF family is envisioned for first-order phase transitions, where standard MCMC has supercritical slowing and latent heat or interface tension are hard to measure precisely.These are proposed applications, not tests reported in this study.Source: §5 · p. 9
Appendix A: Raw numerical data and additional plots
Appendix A collects raw SNF susceptibility data and additional topological-charge histograms. It also identifies the provenance of the susceptibility data shown in Fig. 6.
Appendix A: Raw numerical data and additional plots
- Appendix A collects raw SNF topological-susceptibility data as a function of ncool and susceptibility data in physical units used in Fig. 6.Table A.I reports 10^6 × a^4χ versus ncool for each β.Source: Appendix A · p. 13
- Figure A.1 adds histograms of SNF topological-charge samples measured at the end of out-of-equilibrium trajectories, after periodic boundaries are restored..Source: Appendix A · p. 13, Figure A.1 · p. 13
- Table A.II summarizes the data shown in Fig. 6, whose standard-simulation results come from Ref. [57].The t0 determinations are interpolations of Ref. [33].Source: Appendix A · p. 13