Authors: Gurtej Kanwar, Scott Lawrence, Alessandro Nada, Phiala Shanahan
Organizations: Higgs Centre for Theoretical Physics, The University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK · Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA · Dipartimento di Fisica, Universit`a degli Studi di Torino and INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Turin, Italy · Center for Theoretical Physics – a Leinweber Institute, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · The NSF AI Institute for Artificial Intelligence and Fundamental Interactions
Machine learning (ML) methods have been explored as a means of optimizing lattice field theory calculations, with increasing interest in recent years. This chapter gives an introduction and summary to the use of ML methods to optimize lattice field theory without introducing any uncontrolled approximations. We focus on those directions which are clearly established as of late 2025, marking the 50th anniversary of the beginning of the study of lattice gauge theory. This chapter is a contribution to the online book ``Lattice QCD at 50 years,'' (LQCD@50), edited by Tanmoy Bhattacharya, Maarten Golterman, Rajan Gupta, Laurent Lellouch, and Steve Sharpe.
Figures & tables
Approach
ML application
Area
Optimize existing algorithm
Correlated sampling
Obs. estimation
Tune MG params.
Config. generation, Obs. estimation
Learned interpolators
Obs. estimation
Perfect actions
Action
Transform the problem
Learned-manifold HMC
Config. generation
Learned preconditioners
Solvers & inversion
Table 1: Classification of some ML applications to LQFT based on the part of the computational pipeline that is targeted, and the approach to maintaining exactness. Table is not exhaustive.
Figure 1: Schematic comparison of the sampling cost using a traditional Monte Carlo sampler, such as Hybrid Monte Carlo, versus an optimized flow-based sampler. The up-front cost of training a flow-based sampler is typically larger than the cost of thermalizing a Markov Chain sampler, while the cost per uncorrelated sample is typically smaller for a flow-based sampler. If the number of required samples exceeds the threshold N∗ , the overall cost of sampling is lower when using the flow-based sampling strategy.
Figure 2: Comparison of applying a trivializing flow to improve HMC mixing time (top panel) versus using the same trivializing flow to perform direct importance sampling (bottom panel). The target model density is selected as a simple toy model with two widely separated modes, for which traditional sampling methods suffer from freezing.
Figure 3: Illustrative comparison of the three transport strategies outlined in the main text.
Figure 4: Two views of the signal-to-noise problem, on synthetic data. Left: the correlator on a log-linear plot. The statistical errors grow relative to the signal, although both fall exponentially. Right: an effective mass plot. The smallest mass cannot be effectively determined at early times due to excited-state contamination, and it cannot be determined at late times due to the falling signal-to-noise.
Figure 5: A cartoon of a valid contour deformation. The original domain of integration is the real line, which is then deformed to the curve γ . Together, the two form the boundary of a region Ω (symbolically, ∂Ω=R−γ ) on which the function being integrated is holomorphic.
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
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China. · Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, 200240, China. · State Key Laboratory of Dark Matter Physics, Shanghai Jiao Tong University, Shanghai, 200240, China. +6