# Advances in machine-learning-based sampling motivated by lattice quantum
chromodynamics

Kyle Cranmer  Physics Department, University of Wisconsin-Madison, Madison,
WI, USA  Gurtej Kanwar  Albert Einstein Center for Fundamental Physics,
Institute for Theoretical Physics, University of Bern, Bern, Switzerland
Sébastien Racanière  Google DeepMind, London, UK  Danilo J. Rezende  Google
DeepMind, London, UK  Phiala E. Shanahan  Center for Theoretical Physics,
Massachusetts Institute of Technology, Cambridge, MA, USA  The NSF AI
Institute for Artificial Intelligence and Fundamental Interactions, Cambridge,
MA, USA  e-mail<EMAIL_ADDRESS>
###### Abstract

Sampling from known probability distributions is a ubiquitous task in
computational science, underlying calculations in domains from linguistics to
biology and physics. Generative machine-learning (ML) models have emerged as a
promising tool in this space, building on the success of this approach in
applications such as image, text, and audio generation. Often, however,
generative tasks in scientific domains have unique structures and
features—such as complex symmetries and the requirement of exactness
guarantees—that present both challenges and opportunities for ML. This
Perspective outlines the advances in ML-based sampling motivated by lattice
quantum field theory, in particular for the theory of quantum chromodynamics.
Enabling calculations of the structure and interactions of matter from our
most fundamental understanding of particle physics, lattice quantum
chromodynamics is one of the main consumers of open-science supercomputing
worldwide. The design of ML algorithms for this application faces profound
challenges, including the necessity of scaling custom ML architectures to the
largest supercomputers, but also promises immense benefits, and is spurring a
wave of development in ML-based sampling more broadly. In lattice field
theory, if this approach can realize its early promise it will be a
transformative step towards first-principles physics calculations in particle,
nuclear and condensed matter physics that are intractable with traditional
approaches.

##  1 Introduction

Theoretical nuclear physics has the ironic feature that although the
fundamental laws are well understood, the computations required to make
quantitative, first-principles predictions are in many cases currently
infeasible. The strong nuclear force is fundamentally described by the quantum
field theory known as Quantum Chromodynamics (QCD), which details the dynamics
of constituent particles—quarks and gluons—that arise as excitations of
underlying quantum fields. This theory successfully predicts a wide range of
phenomena that occur at different energy scales, ranging from the high-energy
collisions at the Large Hadron Collider to the properties and interactions of
composite particles such as the proton and neutron, as well as the nuclei they
form. At high energies, the interactions between quarks and gluons are weak,
and accurate QCD calculations can be made using a perturbative expansion,
which is often represented with Feynman diagrams. At the lower energies
relevant for much of nuclear physics, the interactions between quarks and
gluons are strong and the perturbative approach breaks down. In this regime,
quantitative predictions can be achieved through a computational approach
known as lattice QCD, in which the quark and gluon fields are represented on a
discrete spacetime lattice. Many key aspects of nuclear physics can be
computed precisely in this framework. For example, such calculations reveal
how the masses of the proton and neutron arise from the fundamental quarks and
gluons [1], and they have been used to make predictions of the masses of new
composite particles later discovered by experiments at CERN [2, 3, 4].
However, the reach of this approach is limited by its computational cost, and
controlled first-principles QCD calculations of nuclear structure and
reactions, for example, would require a scale of computational resources that
is currently infeasible [5]. Without breakthrough developments, many important
studies will remain impossible even with the world’s next generation of
exascale supercomputers (quintillions ($10^{18}$) of operations per second, or
the equivalent of 50 million laptops working in concert). If the computational
cost of lattice field theory can be greatly reduced, fundamental questions in
particle, nuclear and condensed matter physics will be answered. For example,
first-principles calculations can probe the fine-tunings in nuclear physics
that are deeply important for understanding our existence, by revealing how
sensitive the production of carbon in the Universe via the triple-$\alpha$
process is to the free parameters of the theory, explaining why protons and
neutrons cluster inside nuclei, and elucidating how the lightest elements
formed in the first minutes of the Universe’s existence via Big Bang
nucleosynthesis [6].

Calculations in lattice QCD are cast in the form of statistical averages with
respect to a distribution of quark and gluon field configurations. A major
component of the computational cost of lattice QCD calculations is the
estimation of these averages by Monte Carlo sampling techniques. (Sampling is
one of several computationally-intensive steps in lattice QCD calculations.
Others, such as the inversion of Dirac operators for the calculation of
physical observables, may also be accelerated using machine learning (ML)
approaches [7, 8, 9]). Sampling representative configurations of a system to
quantitatively evaluate its properties is ubiquitous in physics, being used in
fields spanning from ab-initio molecular dynamics and statistical physics to
astrophysics, and many others. However, sampling from the highly-structured,
high-dimensional, and multi-modal distribution of configurations in lattice
QCD presents an extraordinarily difficult computational challenge. This
problem has historically been the impetus for the development of what have
become foundational techniques in computational statistics and high-
performance computing, with far-reaching implications within and beyond
physics. For example, both the classic Metropolis–Hastings Markov chain Monte
Carlo algorithm [10] and Hamiltonian/hybrid Monte Carlo (HMC) [11] were first
developed in the context of theoretical nuclear physics, with the latter
conceived specifically for lattice QCD. Similarly, the IBM Blue Gene series of
supercomputers trace their origins back to the QCDOC (quantum chromodynamics
on a chip) computer built specifically for this particular application [12].

The rapid advance of ML over the past few years has spurred the emergence of a
new class of algorithms that are revolutionizing computing for both science
and industry applications. For example, the extraordinary success of the ML
tool AlphaFold [13] in protein folding took the world of biology by surprise,
redefining the pace of progress in a field where algorithmic developments had
been slow for decades. For lattice QCD, which has historically driven a
virtuous cycle of innovations in scientific computing, these advances promise
a new chapter. In particular, the rise of generative modelling with ML [14,
15] suggests the particular application of sampling algorithms for lattice
QCD. The sampling problem in lattice QCD has several key features that present
both challenges and opportunities to ML. On the one hand, any algorithm must
be asymptotically exact, preventing the direct application of certain
generative ML approaches such as generative adversarial networks or
variational autoencoders (VAEs). A practical challenge is also presented by
the extreme scale of lattice QCD samples used in state-of-the-art
calculations, each of the order of several terabytes at the current time. On
the other hand, the forms of the relevant probability distributions are
exactly known, which can inform the design and training of sampling
architectures. In particular, these distributions are invariant under
complicated and high-dimensional symmetry groups which significantly reduce
the dimensionality and complexity of the problem if they can be incorporated
exactly. Although it has required considerable effort to develop ML models
that incorporate the symmetries of lattice QCD into ML architectures, the
investment has paid dividends in the efficacy of the resulting algorithms.

This Perspective reviews the unique requirements and features of a class of
ML-based sampling strategies that have been recently developed for lattice QCD
applications and places these developments in the broader context of ML for
sampling in scientific domains. Although this endeavour remains in its early
stages, it is already clear that it has considerable potential, not only to
emulate the transformative impact that ML has had in applications such as
AlphaFold [13], but also to spur the advancement of ML itself.

##  2 Lattice QCD and the sampling problem

The lattice method for computing physical observables in quantum field
theories such as QCD proceeds by discretizing space and time onto a four-
dimensional grid (or ‘lattice’), with spacing $a$ between neighbouring points
and a finite volume $V$. In this framework, the fundamental particle degrees
of freedom of the theory— quarks and gluons in QCD—are represented through
‘quantum fields’ that consist of complex numbers, vectors or matrices
associated with the points and edges (or ‘links’) of the lattice. Quantities
of physical interest are then defined by integrals over these field degrees of
freedom, and the continuum, infinite-volume theory is recovered by taking the
limit $a\rightarrow 0$, $V\rightarrow\infty$.

A general physical observable can be defined in terms of quantum ‘operators’
$\mathcal{O}$ and computed as a statistical expectation value[16]:

$\displaystyle\langle\mathcal{O}\rangle=\int\mathcal{D}\Phi\,\mathcal{O}[\Phi]p[\Phi],\text{
where }p[\Phi]=e^{-S[\Phi]}/Z.$ (1)

Here the notation $\int\mathcal{D}\Phi$ schematically indicates integration
over all configurations of the discretized quantum fields collectively denoted
by $\Phi$, and $Z=\int\mathcal{D}\Phi e^{-S[\Phi]}$ is a normalizing constant.
The ‘action’ $S[\Phi]$ encodes the dynamics of the theory by defining the
statistical distribution $p[\Phi]$; in QCD, it describes the fluctuations and
interactions of the quark and gluon fields. The operator $\mathcal{O}$ can be
chosen to study various physical properties of the theory; for example, the
mass of the proton can be calculated using an operator that represents the
interaction of two up quarks and one down quark.

In practice, the integral in equation (1) cannot be computed analytically and
is instead evaluated by Monte Carlo integration, that is, using an ensemble of
$N$ field configurations $\\{\Phi_{1},\dots,\Phi_{N}\\}$ sampled from the
distribution $p[\Phi]$. Physical quantities are then computed as
$\langle\mathcal{O}\rangle\approx\frac{1}{N}\sum_{i=1}^{N}\mathcal{O}[\Phi_{i}]$
with an uncertainty that is systematically improvable by taking $N$ large. The
first step of any lattice field theory calculation is thus a sampling problem.
Although the challenge of generating lattice field configurations is
reminiscent of sampling problems in many other fields, the structure of the
quantum fields, the complicated symmetries of the distribution $p[\Phi]$ and
the sheer scale of the required calculations set this apart as a particularly
difficult computational problem.

###  2.1 Structure and symmetries of field configurations

Figure 1:  Depiction of a single cube within the spacetime lattice of a
lattice QCD calculation. Shown are some elements $U_{\mu}(x)$ of the
discretized gluon field (red), each associated with an edge $(x,x+\hat{\mu})$
from site $x$ to the neighboring site in direction $\mu\in\\{1,2,3,4\\}$, and
an element $\Psi(y)$ of the discretized quark field (blue), associated with a
site $y$. The value of each $U_{\mu}(x)$ is a complex unitary $3\times 3$
matrix with determinant $1$, that is, an SU(3) matrix, and each $\Psi(y)$ is a
$4\times 3$ complex matrix. $a$ is the lattice spacing between neighbouring
points. The fourth dimension of the lattice is suppressed for clarity.

In typical lattice quantum field theories, the discretized quantum fields not
only extend over the spacetime lattice, but also have ‘internal’ degrees of
freedom represented mathematically by a vector or matrix structure at each
point or edge of the lattice. In particular, in QCD the gluon field $U$ is
encoded by $SU(3)$ variables—$3\times 3$ complex unitary, unit-determinant
matrices—on each edge of the lattice, whereas the quark fields $\Psi$ are
encoded by $4\times 3$ complex matrices on each site of the lattice, as shown
in Fig. 1. For QCD, the calculation of a physical observable via equation (1)
can thus be expressed as

$\begin{gathered}\begin{aligned}
\langle\mathcal{O}\rangle&=\frac{1}{Z}\int\mathcal{D}U\mathcal{D}\bar{\Psi}\mathcal{D}\Psi\mathcal{O}[U,\bar{\Psi},\Psi]e^{-S[U,\bar{\Psi},\Psi]}\\\
&=\frac{1}{Z}\int\mathcal{D}U\mathcal{O}^{\prime}[U]e^{-S_{\mathrm{eff}}[U]},\end{aligned}\\\
\text{where }Z=\int\mathcal{D}U\mathcal{D}\bar{\Psi}\mathcal{D}\Psi
e^{-S[U,\bar{\Psi},\Psi]}.\end{gathered}$ (2)

Here the notation $\int\mathcal{D}U$ indicates integration over all values of
the discretised gluon field $U$, whereas the integration
$\int\mathcal{D}\bar{\Psi}\mathcal{D}\Psi$ over all values of the discretized
quark fields are Gaussian integrals that are evaluated analytically, yielding
a modified operator $\mathcal{O}^{\prime}$ and the modified weight
$p[U]=e^{-S_{\mathrm{eff}}[U]}/Z$ over gluon field configurations. (In
particular, the integral $\int\mathcal{D}\bar{\Psi}\mathcal{D}\Psi$ is a
Berezin integral[17] over elements of a Grassmann algebra, which must be
analytically treated to produce an integral amenable to numerical evaluation.)
In practice, auxiliary degrees of freedom known as ‘pseudo-fermions’ [18] are
also typically introduced as stochastic estimators for determinants appearing
in $p[U]=\exp(-S_{\mathrm{eff}}[U])/Z$. State-of-the-art lattice QCD
calculations involve fields of size up to $256^{3}\times 512\approx 8.6\text{
billion}$ lattice sites with quantum fields represented by roughly $50$
degrees of freedom per lattice site (this counting includes four $SU(3)$
matrices for each lattice site, yielding $4\times 8=32$ degrees of freedom, as
well as complex $4\times 3$ matrices with $2\times 3\times 4=24$ degrees of
freedom for each site, arising from the pseudo-fermion fields), meaning that,
in practice, calculations involve Monte Carlo integration over as many as
$10^{12}$ variables.

Symmetries in a lattice field theory manifest as transformations of field
configurations that leave the probability density $p[U]$ and the integration
measure invariant. The action, and hence $p$, is typically invariant under
both discrete geometric symmetries of the hybercubic Euclidean spacetime, such
as discrete translations, rotations and reflections, and under internal
symmetry transformations. For example, one contribution to the lattice QCD
action is given by

$S_{g}[U]=-\frac{\beta}{6}\sum_{x}\sum_{\begin{subarray}{c}\mu=1\\\
\nu=1\end{subarray}}^{4}\operatorname{Re}\operatorname{Tr}[U_{\mu}(x)U_{\nu}(x+\hat{\mu})U^{\dagger}_{\mu}(x+\hat{\nu})U^{\dagger}_{\nu}(x)],$
(3)

where $\beta$ is a parameter of the theory that is related to the lattice
spacing $a$, $x$ is summed over the sites of the discretized lattice, and
$\hat{\mu},\hat{\nu}$ indicate vectors of length $a$ in the $\mu$ and $\nu$
directions, respectively (see Fig. 1). From this expression, it can be seen
how ‘gauge’ symmetry is manifest in QCD, as $p[U]$ is invariant under the
transformation of the gauge field $U$ according to

$U_{\mu}(x)\rightarrow\Omega(x)U_{\mu}(x)\Omega^{\dagger}(x+\hat{\mu})$ (4)

for all possible choices of $\Omega(x)\in\text{SU(3)}$ over all lattice sites.
Because this symmetry is specified by one SU(3)-valued matrix per lattice site
(so eight degrees of freedom per site), the symmetry group may have a
dimension as large as $10^{11}$ in state-of-the-art calculations.

###  2.2 Approaches and challenges to sampling field configurations

Conventionally, the generation of an ensemble of lattice fields distributed
according to $p[\Phi]$ is performed iteratively using a Markov process, in
which a chain of configurations $\\{\Phi_{1},\Phi_{2},\ldots\\}$ is generated
by a sequence of stochastic updates beginning from an initial configuration
$\Phi_{0}$. In particular, the HMC algorithm was first conceived of in the
1980s specifically for this application in lattice field theory [11] and has
since become a mainstay of the computational science community. In this
paradigm, the rapid exploration of the state space is achieved by a directed
evolution from each configuration to a new proposed configuration, which
avoids an inefficient random walk. Exactness of the distribution is guaranteed
by applying the Metropolis–Hastings procedure to accept the proposed
configuration with an appropriate probability [10, 19] (see also the next
section).

Despite the outstanding success of this approach — which remains the workhorse
of lattice field theory — generating ensembles of field configurations is one
of the notable computational costs of first-principles QCD calculations. In
particular, because the approach evolves configurations via a local dynamical
process, increasingly many updates are required to decorrelate samples on
physical length scales as the continuum limit is approached ($a\rightarrow
0$). This is a manifestation of the phenomenon known as ‘critical slowing-
down’ in this context [11, 20]. Simultaneously, the distribution of QCD gauge
fields spans topologically distinct sectors, and Markov-based sampling
algorithms such as HMC can become ‘trapped’ or ‘frozen’ in sectors of fixed
topology.

Any alternative approach to sampling lattice field configurations will need to
satisfy several key requirements in order to be practically viable. Most
importantly, it must be statistically improvable: that is, the true
probability distribution $p[\Phi]$, including the various symmetries that this
distribution respects, must be recovered in the limit of a large number of
samples. Furthermore, the approach must be efficiently scalable to state-of-
the-art lattice field theory studies, which involve field configurations as
large as many terabytes of memory each, with as many as $10^{12}$ degrees of
freedom. Finally, the approach must improve upon the impressive success of the
HMC framework and mitigate the challenges of critical slowing-down and
topological freezing in some regime of physical interest.

Figure 2: Comparison between the sampling tasks of quantum field generation
for lattice quantum chromodynamics and image generation. In addition to
differences in the target and symmetries of the problems, the hierarchy of
degrees of freedom (dof) per sample to number of samples is inverted for
quantum field generation as compared with image generation. The action $S$
encodes the dynamics of the theory by defining the statistical distribution
$p$, $U$ the gluon field, and $Z$ is a normalizing constant. The image on the
right side is reprinted from Kaggle
(https://www.kaggle.com/datasets/vitaliykinakh/stable-imagenet1k) under a
Creative Commons license (CC0 1.0).

##  3 ML for sampling lattice field configurations

An ML approach to sampling lattice field configurations is an appealing
proposition: it offers a new paradigm of algorithms that are optimised
specifically for the task at hand. Enabling radically different approaches to
sampling, ML may mitigate critical slowing-down and other key challenges faced
by traditional Markov-process algorithms such as HMC. Even for cases where HMC
works well, ML may still provide advantages, for example by enabling
embarrassingly parallel, rather than sequential, sampling, or by learning to
approximate a large number of computational steps of traditional algorithms
with fewer operations, as is observed in other fields such as ML
approximations of partial differential equations solvers[21].

However, the application of ML to lattice field theory is not straightforward,
given the previously highlighted features of the lattice field theory problem.
In particular, for the theory of QCD, gauge field samples are collections of
matrices that are constrained to be SU(3) matrices (see Fig. 1), whereas
samples in typical ML domains such as images or natural language models are
represented by vectors of unconstrained real numbers. When considering
generative models based on diffeomorphisms, as discussed below, it is
imperative to ensure that these constraints are satisfied by all
transformations. The associated SU(3) gauge symmetry is also very atypical in
comparison with usual ML applications, although other symmetries such as 4D
translations may be handled by traditional ML methods such as convolutional
neural networks. Moreover, generative models for traditional applications such
as images and language do not require asymptotic guarantees of exactness in
sampling, whereas these are critical in the lattice field theory context.
Finally, the sheer scale of state-of-the-art lattice QCD calculations, both in
terms of the scale of lattice samples and the computational cost required to
manipulate them, presents a challenge to ML approaches. Figure 2 illustrates
these stark contrasts between the lattice field generation problem and other
sampling tasks that have been revolutionized through ML, such as image
generation. Clearly, achieving state-of-the-art sampling performance with new
ML algorithms in the context of lattice field theory will require the
development of new algorithms and innovation in ML.

###  3.1 Classes of generative models for sampling in lattice field theory

ML models designed to (approximately) sample from a target density are known
as generative or probabilistic models. A generative model typically consists
of three components: a space of latent or hidden variables equipped with a
density, a set of observed or target variables, and a parametric map that
transforms points in the latent space into points in the target space.
Optimization is performed on the parametric map so that the density it induces
in the target space approximates the target density. A wide variety of ML-
based generative architectures have been developed over the past decade, with
transformative successes particularly evident in applications to sound/image
data [22, 23, 24, 25] and language data [14, 26, 27, 28, 29, 30, 15].

One notable difference between these applications and the challenge of
sampling field configurations for lattice QCD is that the true distribution
over the space of images, sounds or text is not known, so the model
distribution is learnt from data samples. For lattice field configurations,
not only is the unnormalized target distribution known, but it must be sampled
from with asymptotic guarantees of exactness. This can be achieved with ML
models if they feature tractable likelihoods (the model probability density
can be computed for any given sample); in this case, they can be embedded
inside sampling algorithms with asymptotic guarantees, such as a Markov chain,
as discussed further below.

A tractable likelihood also allows one to optimize an ML model by minimizing a
probability divergence $D(q_{\theta};p)$ between the model probability density
$q_{\theta}[U]$ parameterized by $\theta$ and the known target probability
density $p[U]=e^{-S_{\mathrm{eff}}[U]}/Z$. However, in stark contrast to
typical ML sampling problems, training models for lattice QCD sampling
requires estimating the gradients $\nabla_{\theta}D(q_{\theta};p)$ using only
samples from the model or perhaps only a small number of ‘ground truth’ data
samples. This restricts the family of probability divergences that can be
used; $f$-divergences such as the Kullback–Leibler divergence [31] are
commonly used, as they can be expressed as an expectation value under
$q_{\theta}$, allowing the divergence and its gradient to be estimated from
model samples alone.

Any ML approach to sampling for lattice QCD must also be ‘scalable’ as the
number of lattice sites, $M=V/a^{4}$, is increased. Ideally, its computational
and memory costs should scale linearly or sub-linearly with the number of
lattice sites, which we denote by $O(M)$ below. This applies to all aspects of
the model: drawing samples, evaluating the likelihood $q_{\theta}$, and
evaluating the gradients $\nabla_{\theta}D(q_{\theta};p)$. This consideration
restricts or rules out certain classes of models, as discussed below.

The features of various generative modelling frameworks that could be
considered for the lattice QCD sampling problem are outlined below.

  * •

Latent-variable models such as generative adversarial networks[32] and
VAEs[33, 34] typically have efficient $O(M)$ sampling, but intractable
likelihoods (involving marginalization of latent variables).

  * •

Auto-regressive models [22, 35, 14, 26, 27, 28, 29, 30, 15] typically have
efficient $O(M)$ likelihood evaluation. Sampling can also be achieved with
$O(M)$ cost in principle, but existing implementations are impractically slow.

  * •

Continuous time models include diffusion models [23, 24] defined via
stochastic differential equations and continuous-time normalizing flows[36]
defined via ordinary differential equations. In these models, likelihood
computation requires integrating a scalar ordinary differential equation
defined by the divergence of a vector field (the marginal score function).
This computation typically has computational cost $O(M^{3})$ unless additional
structure is forced on to the model[37].

  * •

Discrete time normalizing flow models[38, 39, 40, 41, 42, 43] remain as good
candidate models. In a discrete flow, the generative process maps a latent
vector $z$ (a field configuration in the lattice field theory context) sampled
from a base density into the target density via the composition of a series of
parametric diffeomorphisms $F_{1}$, $F_{2}$, …, $F_{n}$. If $z$ is sampled
with density $r(z)$, then a flow sample $x=F(z)$ has known density
$q(x)=r(z)\left|\det{\partial F/\partial z}\right|^{-1}$, where
$F=F_{1}\circ\dots\circ F_{n}$ is the composed diffeomorphism. By restricting
to $F_{i}$ for which $\partial F_{i}/\partial z$ is a triangular matrix, the
cost of $\left|\det{\partial F/\partial z}\right|$ is only $O(M)$ (ref. 38).

These models, however, also have intrinsic limitations that must be worked
around, such as topology preservation of the diffeomorphism [44] and
difficulty in modelling tail-behaviour of a target density if the tails are
not already in the base density [45].

One can also consider sampling in an augmented space, where the data space is
augmented with an additional set of auxiliary or latent variables. In this
setting, it may be viable to reconsider VAEs [33, 34] or VAE–flow hybrids
[46], but currently there are no results demonstrating these methods perform
well compared to models working directly on the data space for lattice QCD.

###  3.2 Methods to guarantee asymptotic exactness

Several mechanisms have been proposed to combine generative models with Markov
chain Monte Carlo and importance-sampling algorithms in order to inherit their
asymptotic convergence guarantees. One of the simplest mechanisms is neural
importance sampling, in which a model density $q_{\theta}(x)\approx p(x)$ is
used to evaluate expectations under the target $p$ via
$\mathbb{E}_{p}[\mathcal{O}(x)]=\mathbb{E}_{q_{\theta}}[\frac{p(x)}{q_{\theta}(x)}\mathcal{O}(x)]$
(ref. 47). An appealing alternative is to incorporate generative models into
an asymptotically exact Markov process, which allows existing analysis
techniques to be used or existing Markov chain updates to be combined with the
ML sampling approach. Generally, the Metropolis–Hastings algorithm uses an
ergodic transition kernel $K(x^{\prime}|x)$ to propose Markov chain updates
$x\rightarrow x^{\prime}$ which are accepted with probability

$p_{\mathrm{acc}}(x^{\prime}|x)=\min\left(1,\frac{K(x|x^{\prime})p(x^{\prime})}{K(x^{\prime}|x)p(x)}\right)$
(5)

to ensure that the asymptotic equilibrium distribution is the desired target
distribution [10, 19]. Although this method is guaranteed to converge [48],
the speed of convergence depends on the target density and the choice of
transition kernel $K(x^{\prime}|x)$.

ML models can be combined with the Metropolis–Hastings approach by using
generative models to construct the kernel $K(x^{\prime}|x)$. A direct approach
is to use the trained model $q_{\theta}(x)\approx p(x)$ to produce independent
and identically distributed (iid) proposal samples,
$K(x^{\prime}|x)=q_{\theta}(x^{\prime})$, with the convergence rate determined
by the quality of the model approximation to the target distribution. More
advanced techniques include neural transport Monte Carlo[49, 50], where Markov
chain Monte Carlo is performed in the latent space; learned Monte Carlo
proposals[51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64], where the
goal is to directly learn the kernel $K(x^{\prime}|x)$; learned sequential
Monte Carlo sampling [65, 66, 67, 46], which combines deterministic flows with
sequential Monte Carlo annealing techniques; Monte Carlo variance minimization
[47, 68, 69]; and stochastic normalizing flows [46], which interleave
deterministic transforms with latent-variable VAE-like components and Markov
chain Monte Carlo transforms. HMC is a particularly successful family of
kernels and can also be combined with learnt components. In the presence of
pseudo-fermions, simulating the Hamiltonian dynamics for HMC requires an
expensive computation of ‘force terms’, which is another area where
acceleration may be possible using ML [70, 71, 72, 73, 62].

###  3.3 Incorporating manifold constraints and gauge symmetry in ML models

When the target density for sampling features an exact or approximate
symmetry, breaking that symmetry in a sampling algorithm will result in
computational inefficiencies. In particular, continuous exact symmetries
naturally reduce the effective dimensionality of the target distribution, such
that incorporating them directly reduces the difficulty of modelling that
target. In addition, training a model with the symmetry explicitly encoded
modifies the structure of the loss landscape in a way that may make training
feasible in cases where attempting to approximately ‘learn’ the symmetry may
not be. Moreover, if guarantees of exactness are required, any remaining
symmetry-breaking after training will result in additional costs to correct
that breaking via the approaches discussed in the previous subsection. It is
thus often advantageous, and in some cases critical, to incorporate symmetries
into ML architectures and/or training approaches.

Modern ML offers many techniques that seek to take advantage of known
symmetries. The simplest such method is data augmentation[74], where the
available training set is ‘augmented’ with randomly transformed input/output
pairs. Another technique consists of explicitly adding a term to the
optimization target that encourages equivariance or invariance with respect to
a group of transformations[75, 76]. Both approaches only serve to assist the
training, and symmetries still have to be learnt by the model architecture.
Alternatively, it is also possible to construct architectures such that they
respect known symmetries by construction; standard convolutional neural
networks, for example, are equivariant maps with respect to translations. The
most commonly studied symmetries are finite groups[77] and SE(3), the group of
isometries of 3D Euclidean space[78, 79]. Naturally, the larger the symmetry
group, the harder it is to learn the symmetries, either via data augmentation
or via additional optimization targets. In the case of lattice QCD, the gauge
symmetry has a prohibitively large dimension scaling with the number of
lattice sites, hence it is likely essential to build gauge-equivariant and
invariant neural networks for this application. For example, with current
architectures and training approaches it has been demonstrated that it is
essential to exactly incorporate gauge symmetry [80], but not translations and
hybercubic transformations [81], for successful training of flow-based
sampling algorithms for lattice field theory.

Building generative models that exactly incorporate the symmetry constraints
of lattice QCD is a non-trivial task that has required the introduction of
several new ML models to treat both gauge and pseudo-fermionic degrees of
freedom[82, 81, 80, 83, 84, 85, 86, 87, 88, 89]. This approach relies on the
observation that starting from a base distribution that is gauge-invariant
(such as the Haar measure on $\text{SU}(N)$) and applying a gauge-equivariant
diffeomorphism to this base density yields a new density that is also gauge-
invariant[38, 90, 76, 83]. One can reduce the problem of building gauge-
equivariant diffeomorphisms on the gauge degrees of freedom situated on the
edges of a lattice to the problem of building matrix-conjugation-equivariant
diffeomorphisms[81, 80] on SU(3), which is a simpler problem. As described in
equation (4), a gauge symmetry transformation $T_{\Omega}$ is parameterized by
a field of SU(3) variables $\Omega(x)$ and acts on an edge, or ‘link’,
variable as
$T_{\Omega}U_{\mu}(x)=\Omega(x)U_{\mu}(x)\Omega^{\dagger}(x+\hat{\mu})$. As a
consequence, a product of link variables along a closed loop $\Lambda(x)$
starting and ending at a point $x$ transforms as
$\Lambda(x)\rightarrow\Omega(x)\Lambda(x)\Omega^{\dagger}(x)$. Mathematically
this is referred to as a matrix-conjugation, or adjoint transformation. If
$U_{\mu}(x)$ is a link, $\Gamma(x,x+\hat{\mu})$ is a product of links along an
open path from $x$ to $x+\hat{\mu}$ that does not contain $U_{\mu}(x)$, and
$g:\text{SU(3)}\rightarrow\text{SU(3)}$ is a conjugation-equivariant
diffeomorphism, then a gauge-equivariant diffeomorphism $f$ can be constructed
as

$f(U_{\mu}(x))=g(U_{\mu}(x)\Gamma^{\dagger}(x,x+\hat{\mu}))\Gamma(x,x+\hat{\mu}),$
(6)

as shown in Fig. 3. The diffeomorphism $f$ then manifestly satisfies gauge
equivariance, $f(T_{\Omega}U)=T_{\Omega}f(U)$. Building gauge-equivariant
diffeomorphisms for pseudo-fermion degrees of freedom is also possible using
extensions of this approach based on parallel transport[91].

Figure 3: Illustration of a gauge-equivariant transformation layer [81, 80].
Here, a parametric gauge-equivariant diffeomorphism is constructed from a
diffeomorphism $g$ satisfying $g(\Omega U\Omega^{\dagger})=\Omega
g(U)\Omega^{\dagger}$ (matrix-conjugation equivariance). To update the matrix-
valued link $U_{\mu}(x)$ in red, this transformation first updates the
plaquette $U_{\mu}(x)\Gamma^{\dagger}(x,x+\hat{\mu})$ containing that link,
before ‘pushing’ that update onto the link by assigning
$U^{\prime}_{\mu}(x)=g(U_{\mu}(x)\Gamma^{\dagger}(x,x+\hat{\mu}))\Gamma(x,x+\hat{\mu})$.
The output link $U^{\prime}_{\mu}(x)$ transforms appropriately under gauge
transformations when $U_{\mu}(x)$ and $\Gamma(x,x+\hat{\mu})$ are transformed.
See the main text for further details.

###  3.4 A roadmap for ML-based sampling in lattice QCD

The great potential of ML-based sampling for lattice field theories has
inspired rapid developments that have already demonstrated profound successes.
Following early applications of flow-based sampling to field theories other
than QCD [88, 80], the approach was developed for theories in two spacetime
dimensions, specifically for SU(3) gauge fields without dynamical quark
(fermionic) degrees of freedom [81] and for U(1) gauge fields with dynamical
fermions [86]. Combining these advances enabled the first application of flow
models to sampling QCD in 4D [92], albeit with small space-time volumes. ML-
accelerated updating schemes have been developed, again for small volumes and
with the SU(2) gauge group instead of SU(3) (ref. 62), and continuous-time
models inspired by previous work in the lattice field theory community [93,
94] have been applied to simple lattice field theories [95] and both U(1)
gauge theory [82] and SU(3) gauge theory [96] in 2D. These approaches have had
astounding success; Fig. 4 illustrates the advantages of flow-based sampling
in one particular toy theory, but the conclusion that ML-accelerated sampling
schemes can overcome the critical slowing-down and topological freezing
challenges faced by HMC has been clear and universal. It is important to
emphasize that this success includes theories with fermionic degrees of
freedom, where ML-accelerated sampling schemes have been developed to
integrate with the usual approach of pseudo-fermions [91, 97, 92]. However,
the crucial aspect missing in all applications so far is a demonstration of
the effectiveness of ML techniques at the scale of state-of-the-art lattice
QCD calculations in nuclear and particle physics.

Figure 4: Demonstration of the advantages of flow-based sampling in a U(1)
lattice gauge theory in two spacetime dimensions [80]. The inset shows the
rapid mixing of topological charge $Q$ when sampling with normalizing flows,
compared with Hamiltonian/hybrid Monte Carlo (HMC) and Heat Bath (HB)
algorithms for the action defined by $\beta=7$ (see equation (3) for the
definition of the analogous parameter in quantum chromodynamics). The main
graph shows the asymptotically improved scaling of $\tau^{\mathrm{int}}_{Q}$
towards the continuum limit $\beta\rightarrow\infty$, where
$\tau^{\mathrm{int}}_{Q}$ is the ‘integrated autocorrelation time’ of the
topological charge, which is a measure of cost in Markov-process sampling and
can be interpreted here as a metric for critical slowing-down. Reproduced with
permission from ref. 80, APS.

We expect that not only will we soon see applications of ML-accelerated
sampling to lattice field theory at scale, but that running at scale is a key
ingredient necessary to realize the full potential of ML in this context. In
particular, we anticipate that the first impact of ML for this application
will be that, once the potentially high cost of training is paid, ML-based
sampling will be orders of magnitudes faster than traditional HMC, mitigating
critical slowing-down, overcoming topological freezing, and opening the door
to a sampling regime where this training cost can be efficiently amortized, as
depicted in Fig. 5. At precisely which scale this advantage will be reached is
not yet clear; the computational cost of training ML models in this context
may vary by orders of magnitude between different architectures and training
approaches [98]. As the optimal approach to model parameterization and
training can depend sensitively on the number of samples which are ultimately
required, the balance of training and sampling costs is highly problem-
dependent, and the regime in which flow-based sampling outperforms HMC for
lattice QCD applications will depend on precisely how the flow models are used
(and reused). It is already evident, however, that achieving this paradigm of
efficient ML-accelerated sampling will require considerable investment; it is
clear that in the field of generative ML as a whole, the substantial progress
in text [14, 26, 27, 28, 29, 30, 15] and image modelling [22, 23, 24, 25] has
required pushing the boundaries of model size. So far, the generative ML
experiments for lattice field theory are of comparatively small scale, despite
the target scale of the problem itself being comparable to, or even larger
than, applications in these domains. Success will thus likely require model
scales, and corresponding investments in upfront training, that will
constitute a change in paradigm of computational resource use for the
theoretical physics community. Nevertheless, as a fundamentally structured
problem, we anticipate that scaling the custom ML solutions developed for
lattice field theory to large models will pay dividends and that lattice QCD
will join the list of scientific problems which have seen significant impacts
from ML at state-of-the-art scale, such as low-dimension Bayesian parameter
inference for astrophysics [99], quantum Monte Carlo [100] and protein-folding
[13].

Beyond the anticipated impacts of mitigating sampling challenges such as
critical slowing-down and topological freezing, ML models have the potential
to catalyse other paradigm shifts in lattice field theory. For example, they
naturally offer new opportunities for community resource sharing. Ensembles of
lattice field configurations are large enough in size (petabytes for state-of-
the-art ensembles) that they can not be easily shared, and massive investments
in tape resources are made to store them. In contrast, even the largest ML
models only contain a few terabytes of parameters. These can easily be shared,
allowing research groups around the globe to efficiently generate their own
configurations or reproduce ensembles from a known seed, in both cases
capitalizing on community-owned pre-trained models. Another important
opportunity is that ML-based samplers can be conditioned on various parameters
of the theory, from the lattice spacing and volume to physical parameters such
as the strength of coupling of the fundamental particles of the theory. The
potential to generate ‘correlated’ sets of samples at different parameters,
interpolated, or even extrapolated[101, 102], from the parameters used during
training, is qualitatively distinct from what is possible using traditional
sampling algorithms such as HMC (in principle this is also possible for pre-
selected parameter sets, using approaches such as parallel tempering), and
offers new parameter extrapolation methods. Ultimately, one could even imagine
more general ways of conditioning these models (for example, on a symbolic
description of the target action), enabling new approaches such as direct
measurements of the effect of modifying the action on physical observables. As
such, ML techniques hold the promise of redefining conventional wisdom in this
field, with implications that have yet to be fully explored.

Figure 5: Sketch of the upfront and sampling costs of hybrid Monte Carlo
compared with flow-based models. Hybrid Monte Carlo (HMC) shown in blue; flow-
based sampling in red. Whereas machine-learning models require a potentially
large upfront computational cost for training, they offer the hope of
efficient sampling, such that this training cost can be amortized. That is,
for sufficiently many samples, flow-based sampling that is more efficient than
HMC (for example, as in Fig. 4) can outperform HMC in terms of total cost. The
minimum number of samples for which flow-based sampling more efficient than
HMC is denoted by $N^{*}$.

##  4 Outlook

As ML continues to evolve in scope and complexity, its applications in science
are being driven into two broad categories: those that can adapt existing ML
technologies (usually developed to model images, sound and text) and those
that demand ground-up development and inspire innovation. Lattice field theory
is becoming established as a prime example of the latter, being simultaneously
an important science application in which algorithmic acceleration will have
wide-reaching implications for fundamental physics and a massive computational
challenge of a scope and scale that has driven advances in computation and
algorithms for decades.

In particular, the challenge of sampling lattice field configurations for
nuclear and particle physics calculations is a definitive proving-ground for
generative ML models in science. With strict requirements of asymptotic
exactness, and an ultimate scale at which each sample is several terabytes in
size, it is clear that symmetries, structure and domain knowledge must be
incorporated into ML architectures designed for this task. In that regard, the
application of ML to sampling in lattice field theory is a key exemplar
informing the debate around the value of engineering and incorporating domain
or expert knowledge that is currently underway in the ML community: whereas
ardent supporters of modern deep learning often argue against the long-term
value of incorporating such knowledge (the ‘bitter lesson’ theory [103],
advocated by Richard Sutton), in the physical sciences, and in particular in
theoretical physics calculations, there are often precise mathematical
formulations of domain knowledge that not only make little sense to ignore,
but are intractable to learn from data. Adding to the complexity of this
engineering and design challenge, existing algorithmic benchmarks for sampling
in lattice field theory are extremely high; new ML samplers must compete
against well-established algorithms that have been optimized in co-design with
high-performance computing systems for more than four decades.

As such, ML for lattice field theory is not only a benchmark for the
application of ML in science, but it is also an endeavour with paradigm-
shifting potential for physics. If the success already achieved in sampling
field theories at toy scales can be mimicked at state-of-the-art scales, it
will transform the computational landscape of a field that is one of the
largest consumers of open-science supercomputing (computing available to
public scientific applications) worldwide, with impacts across particle,
nuclear and condensed matter physics and beyond. On the ML side, it will be a
flagship example of the power of sophisticated domain-specific customization
and engineering to achieve transformative impact in computational science.

## Acknowledgements

We thank W. Detmold and R. D. Young for comments on the manuscript. P.E.S. was
supported in part by the U.S. Department of Energy, Office of Science, Office
of Nuclear Physics, under grant Contract Number DE-SC0011090 and by Early
Career Award DE-SC0021006, by a NEC research award, and by the Carl G. and
Shirley Sontheimer Research Fund. G.K. was supported by funding from the
Schweizerischer Nationalfonds (grant agreement no. 200020_200424).

## Author contributions

The authors contributed equally to all aspects of the article.

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