Lattice Field Theory

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16 papers6 papers in the last four weeks, with none the four weeks before. 0.0% of all new papers.

Oct 14, 2026

Ergodic topology sampling toward the continuum with Stochastic Normalizing Flows

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.

Claudio Bonanno, Andrea Bulgarelli, Elia Cellini +4

Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstraße 5, CH-3012 Bern, Switzerland · Transdisciplinary Research Area “Building Blocks of Matter and Fundamental Interactions” (TRA Matter) and Helmholtz Institute for Radiation and Nuclear Physics (HISKP), University of Bonn, Nussallee 14-16, 53115 Bonn, Germany · Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, United Kingdom +3

Lattice Field Theory High-Energy Physics Normalizing Flows

Oct 13, 2026

Data Free Training of Diffusion Models for Lattice Field Theories

Sampling equilibrium field configurations remains a major computational bottleneck in lattice field theory, motivating the development of generative samplers that can produce independent configurations. Diffusion models are promising for this purpose, but their standard denoising score matching objective requires samples from the target distribution, seemingly reintroducing the original sampling problem. Here, we show that diffusion models can instead be trained from their own generated samples using the target score identity, provided access to the unnormalized probability density. Importance-weighted formulations of this identity yield a consistent target throughout training, while unweighted variants define a self-consistent fixed-point training scheme. We benchmark the different approaches in two-dimensional

φ^4

theory and identify effective choices that enable efficient training and sampling across the symmetric, critical, and broken phases. Ultimately, our data-free training approach achieves comparable performance to training on HMC reference data.

Dennis Grinwald, Ankur Singha, Khaled Kahouli +2

Berlin Institute for the Foundations of Learning and Data (BIFOLD), Germany · Technische Universität Berlin, Germany · Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, 15738 Zeuthen, Germany +2

Lattice Field Theory Diffusion Model Sampling Diffusion Models Score Matching High-Energy Physics

Oct 8, 2026

Machine Learning Meets High-Energy Nuclear Physics: From Pattern Recognition to Physics-Integrated Discovery

Machine learning (ML) in high-energy nuclear physics (HENP) is entering a new stage in which physical knowledge is incorporated more directly into data analysis, simulation, and physics inference. This mini-review focuses on developments that have matured in the past several years. Whereas earlier applications emphasized event classification, pattern recognition, and surrogate models for selected observables, recent work has moved toward physics-integrated workflows: calibrated Bayesian extraction of QCD matter properties, dense-matter equation-of-state inference from heavy-ion and neutron-star data, generative event modeling, neural unfolding of weak physical signals, differentiable inverse solvers, gauge-equivariant and diffusion-based lattice-field samplers, and neural reconstruction of model functions in holographic QCD. We survey recent applications of ML in heavy-ion collisions, neutron-star physics, lattice QFT, and holographic or continuum QCD. The emphasis is not on ML architectures alone, but on how they enter concrete physics workflows, how physical constraints such as symmetries, conservation laws, causality, thermodynamic stability, and topology are imposed, and how uncertainty quantification and validation determine whether an AI-assisted result can support a reliable physics conclusion.

Xun Chen, Weiyao Ke, Yu-Gang Ma +2

School of Nuclear Science and Technology, University of South China, Hengyang 421001, China · INFN — Istituto Nazionale di Fisica Nucleare — Sezione di Bari, Via Orabona 4, 70125 Bari, Italy · Key Laboratory of Quark and Lepton Physics (MOE) & Institute of Particle Physics, Central China Normal University, Wuhan, Hubei 430079, China +6

Lattice Field Theory High-Energy Physics Physics-Informed ML Scientific ML Physics-Informed Generative Modeling

Oct 6, 2026

Sampling SU(N) gauge theory on a 2D lattice from independent plaquettes via holonomies and corner reweighting

In the Wilson formulation of lattice gauge theory, the fundamental degrees of freedom are group-valued link variables, while the action is a sum over the trace of the plaquettes, the smallest Wilson loops. For generative sampling methods such as normalizing flows, this poses a challenge: the distribution of an individual plaquette is easy to model, but mapping sampled plaquettes to the links is the obstruction. We explore a statistical way around this in two dimensions for the

\mathrm{SU}(N)

gauge theory with the Wilson plaquette action. The action can be written in terms of holonomy variables, which determine the links once a consistency condition at the four corners of an extended lattice is satisfied. Using a boundary condition that only affects the holonomies, we trade this consistency condition for a conditional sampling problem, which reduces to sampling

X,Y\in\mathrm{SU}(N)

given the group commutator

Z=XYX^\dagger Y^\dagger

, where

Z\in\mathrm{SU}(N)

depends on the four corners. Plaquettes are sampled independently, and the resulting configurations carry weights due to the additional conditional sampling. We obtain the density of

Z

, which determines these weights, in closed form for

\mathrm{SU}(2)

and

\mathrm{SU}(3)

. A normalizing flow models the single-plaquette distribution for

\mathrm{SU}(2)

and

\mathrm{SU}(3)

; for the group-commutator problem we use a closed-form sampler for

\mathrm{SU}(2)

and a trained normalizing flow for

\mathrm{SU}(3)

. In our tests on

2\times2

and

32\times32

lattices at two couplings each, per-plaquette acceptance rates exceed

99\%

and the effective sample size of the final weights is moderate, typically above one half.

Javad Komijani

Lattice Field Theory High-Energy Physics Importance Sampling

Sep 26, 2026

SIMANF: Sample Free Learning of Unnormalized Distributions via Simulated Annealing in Normalizing Flows

Efficiently learning and sampling from high dimensional, multimodal unnormalized distributions without target samples remains a challenging problem. Although normalizing flows can generate samples efficiently, training based on the reverse KL divergence using only the unnormalized target density may suffer from mode collapse. We introduce SIMANF, a sample free framework that integrates simulated annealing with normalizing flows. SIMANF progressively transforms the target distribution from a smooth initial form to the original target distribution and trains the flow sequentially across these stages. By transferring the learned representation between stages, the method promotes mode coverage while progressively capturing finer features of the target distribution. Following annealing, a final refinement stage combines the reverse KL divergence with an importance weighted forward KL objective using samples generated by the flow. SIMANF requires no target samples during training and uses only the unnormalized density. We demonstrate its effectiveness on Many-Well distributions and high dimensional Scalar Phi4 lattice field theory distribution.

Vikas Kanaujia

Mahindra University Department of Electrical and Computer Engineering Hyderabad, Telangana, India

Lattice Field Theory Sampling from Unnormalized Distributions Normalizing Flows

Sep 24, 2026

Accurate Sampling from Diffusion Models

A new proposal called DM-SMC (Diffusion Model - Sequential Monte Carlo) is investigated, which samples ensembles defined in terms of an action, using diffusion models trained on samples from the ensemble. The SMC setup allows for accurate sampling in spite of an approximate diffusion model and the finite stepsize used in the numerical solution of the stochastic process. Improved update strategies are also investigated. Results are presented for a

Z_2

symmetric scalar field theory in 2 dimensions near its 2nd order phase transition.

Dénes Sexty

Institute of Physics, NAWI Graz, University of Graz, Universitätsplatz 5, 8010 Graz, Austria

Lattice Field Theory Diffusion Model Sampling High-Energy Physics

Jul 17, 2026

Neural Non-Equilibrium Hamiltonian Monte Carlo for Corrected Boltzmann Sampling

Learned dynamical proposals can generate configurations without providing a tractable endpoint density. Nonequilibrium path probabilities offer a way to correct such proposals, but correction alone does not determine their ability to connect separated regions. We introduce Neural Non-Equilibrium Hamiltonian Monte Carlo (NHMC), which combines conditional momentum distributions with reversible, volume preserving dynamics. The forward--reverse path ratio gives the work used for training, importance weighting, normalizer estimation, and Metropolis correction. We then construct a configuration space round-trip kernel whose reverse and forward paths share an intermediate configuration. Conditional on that configuration, its path record update is independence Metropolis--Hastings. We compare its stationary inter-region flow with the flow obtained under exact conditional matching and bound their difference by the conditional path mismatch. This separates errors in the conditional proposal from dependence between the endpoint region and the intermediate configuration. Many well calculations test normalizers and mode probabilities; a controlled lattice

φ^4

calculation compares ideal and accepted inter-sector flows and measures sector relaxation. In two dimensional

U(1)

,

\mathrm{SU}(2)

, and

\mathrm{SU}(3)

gauge theories, corrected path weight estimates reproduce finite volume Wilson loops;

U(1)

topological susceptibilities are also compared with analytic predictions across several lattice volumes.

Moxian Qian

Helmholtz Institute Mainz Johannes Gutenberg University Mainz Institute of Molecular Biology (IMB) Mainz, Germany

Markov Chain Monte Carlo Lattice Field Theory Boltzmann Sampling

Jul 8, 2026

Weight-Space Physics: Interpretable Hypernetworks for Lattice Quantum Field Theories

Lattice field theory is the workhorse of non-perturbative physics, used to simulate phenomena from the strong nuclear force to critical phenomena in materials. Its Boltzmann distributions are parametrized analytically by coupling constants, but these bare parameters are weak predictors of observables -- extracting physics typically requires extensive simulation. While normalizing flows have emerged as effective samplers at fixed couplings, it remains difficult to interpret what these networks have learned. This raises a natural question: can the physics be read off directly from the flow network parameters themselves, and can those parameters be generated for unseen theories? We propose lattice field theory as a testbed for neural network interpretability: because the target physics is qualitatively well-understood and smoothly varying, it provides ideal synthetic data with known ground truth. To this end, we introduce JEPAWG, a Joint-Embedding Predictive Architecture-based Weight Generator that maps couplings directly to flow weights via a learned latent space. On a scalar theory at lattices of size

6^2

to

11^2

, the JEPAWG latent space recovers the correct intrinsic dimension of the underlying manifold, locates the phase transition, and encodes a finite-size shift aligned with the 2D Ising exponent

ν\approx 1

, allowing us to uncover physical structure by studying the network weights alone. This suggests the fascinating idea of treating the network weights as a new type of physical observable. As a generator, JEPAWG also interpolates and extrapolates to unseen couplings effectively and remains robust to weight-space discontinuities introduced by multi-seed training data, outperforming PCA, AE, and VAE baselines.

Tobias Göbel, Julian R. Ebelt, Zier Mensch +2

Institute of Physics, University of Amsterdam, Amsterdam, Netherlands · Department of Physics, National Taiwan University, Taipei, Taiwan · Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA +2

Phase Transitions Neural Network Interpretability Hypernetworks Weight-Space Analysis Lattice Field Theory

Jun 25, 2026

Sampling the Schwinger Model with Gauge-Equivariant Diffusion

We present a first study of a diffusion-based approach to accelerated sampling of the

N_f = 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

Diffusion Sampling Generative Modeling High-Energy Physics Score-Based Generative Modeling Lattice Field Theory

Jun 14, 2026

Learning the generating functional for variance reduction in lattice QCD

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

High-Energy Physics Variance Reduction Lattice Field Theory

May 11, 2026

Operator-Guided Model Reduction for Generative Sampling in Lattice Field Theory

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

φ^4

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

Flow Matching Markov Chain Monte Carlo Normalizing Flows High-Energy Physics Lattice Field Theory

May 7, 2026

Diffusion model for SU(N) gauge theories

Implicit score matching provides a computationally efficient approach for training diffusion models and generating high-quality samples from complex distributions. In this work, we develop a score-matching framework for SU(N) lattice gauge theories, which can be extended to other Lie groups. We apply the method to SU(3) gauge configurations with the Wilson gauge action in two and four dimensions and assess the quality of the generated samples by comparison with Hybrid Monte Carlo (HMC) simulations. We show that the diffusion models can be successfully trained and applied for sampling the Wilson gauge action. For large values of inverse coupling, accurate reverse-time integration requires predictor-corrector schemes, for which we introduce a corrector based on Hamiltonian molecular dynamics. While the corrector significantly improves sampling quality, it also increases the computational cost. We outline several strategies for improving sampling efficiency.

Javad Komijani, Marina K. Marinkovic, Lara Turgut

Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

Score Matching Diffusion Sampling High-Energy Physics Diffusion Models Lattice Field Theory

Apr 22, 2026

Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories

Local gauge symmetry underlies fundamental interactions and strongly correlated quantum matter, yet existing machine-learning approaches lack a general, principled framework for learning under site-dependent symmetries, particularly for intrinsically nonlocal observables. Here we introduce a gauge-equivariant graph neural network that embeds non-Abelian symmetry directly into message passing via matrix-valued, gauge-covariant features and symmetry-compatible updates, extending equivariant learning from global to fully local symmetries. In this formulation, message passing implements gauge-covariant transport across the lattice, allowing nonlocal correlations and loop-like structures to emerge naturally from local operations. We validate the approach across pure gauge, gauge-matter, and dynamical regimes, establishing gauge-equivariant message passing as a general paradigm for learning in systems governed by local symmetry.

Ali Rayat, Yaohang Li, Gia-Wei Chern

Department of Physics, University of Virginia, Charlottesville, VA 22904, USA · Department of Computer Science, Old Dominion University, Norfolk, VA 22904, USA.

Message Passing Neural Networks High-Energy Physics Equivariant GNNs Lattice Field Theory

Apr 14, 2026

Machine learning for four-dimensional SU(3) lattice gauge theories

In this review I summarize how machine learning can be used in lattice gauge theory simulations and what ap-proaches are currently available to improve the sampling of gauge field configurations, with a focus on applications in four-dimensional SU(3) gauge theories. These include approaches based on generative machine-learning models such as (stochastic) normalizing flows and diffusion processes, and an approach based on renormalization group (RG) transformations, more specifically the machine learning of RG-improved gauge actions using gauge-equivariant convolutional neural networks. In particular, I present scaling results for a machine-learned fixed-point action in four-dimensional SU(3) gauge theory towards the continuum limit. The results include observables based on the classically perfect gradient-flow scales, which are free of tree-level lattice artefacts to all orders, and quantities related to the static potential and the deconfinement transition.

Urs Wenger

Institute for Theoretical Physics Albert Einstein Center for Fundamental Physics University of Bern CH-3012 Bern Switzerland

Lattice Field Theory High-Energy Physics

Mar 3, 2026

Variance reduction in lattice QCD observables via normalizing flows

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

Lattice Field Theory Variance Reduction High-Energy Physics Normalizing Flows

Feb 27, 2026

A Monte Carlo estimator of flow fields for sampling and noise problems

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

Lattice Field Theory High-Energy Physics Flow Map Learning