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. 

pdf/advances-in-machine-learning-based-sampling-motivated-by-4ijktsjpks.pdf

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

Abstract Recent applications of machine-learned normalizing flows to sampling in lattice field theory suggest that such methods may be able to mitigate critical slowing down and topological freezing. However, these demonstrations have been at the scale of toy models, and it remains to be determined whether they can be applied to state-of-the-art lattice quantum chromodynamics calculations. Assessing the viability of sampling algorithms for lattice field theory at scale has traditionally been accomplished using simple cost scaling laws, but as we discuss in this work, their utility is limited for flow-based approaches. We conclude that flow-based approaches to sampling are better thought of as a broad family of algorithms with different scaling properties, and that scalability must be assessed experimentally. 

pdf/aspects-of-scaling-and-scalability-for-flow-based-sampling-4uqj8yoiox.pdf

Aspects of scaling and scalability for flow-based sampling of lattice QCD

We study the consequences of mode-collapse of normalizing flows in the context of lattice field theory. Normalizing flows allow for independent sampling. For this reason, it is hoped that they can avoid the tunneling problem of local-update Markov Chain Monte Carlo algorithms for multimodal distributions. In this work, we first point out that the tunneling problem is also present for normalizing flows but is shifted from the sampling to the algorithm's training phase. Specifically, normalizing flows often suffer from mode-collapse for which the training process assigns vanishingly low probability mass to relevant modes of the physical distribution. This may result in a significant bias when the flow is used as a sampler in a Markov-Chain or with importance sampling. We propose a metric to quantify the degree of mode-collapse and derive a bound on the resulting bias. Furthermore, we propose various mitigation strategies in particular in the context of estimating thermodynamic observables, such as the free energy. 

pdf/detecting-and-mitigating-mode-collapse-for-flow-based-2q6cma33np.pdf

Detecting and mitigating mode-collapse for flow-based sampling of lattice field theories

In recent years, machine learning has emerged as a powerful computational tool and novel problem-solving perspective for physics, offering new avenues for studying strongly interacting QCD matter properties under extreme conditions. This review article aims to provide an overview of the current state of this intersection of fields, focusing on the application of machine learning to theoretical studies in high energy nuclear physics. It covers diverse aspects, including heavy ion collisions, lattice field theory, and neutron stars, and discuss how machine learning can be used to explore and facilitate the physics goals of understanding QCD matter. The review also provides a commonality overview from a methodology perspective, from data-driven perspective to physics-driven perspective. We conclude by discussing the challenges and future prospects of machine learning applications in high energy nuclear physics, also underscoring the importance of incorporating physics priors into the purely data-driven learning toolbox. This review highlights the critical role of machine learning as a valuable computational paradigm for advancing physics exploration in high energy nuclear physics. 

/pdf/exploring-qcd-matter-in-extreme-conditions-with-machine-1itzhvl5.pdf

Exploring QCD matter in extreme conditions with Machine Learning

Machine-learned normalizing flows can be used in the context of lattice quantum field theory to generate statistically correlated ensembles of lattice gauge fields at different action parameters. This work demonstrates how these correlations can be exploited for variance reduction in the computation of observables. Three different proof-of-concept applications are demonstrated using a novel residual flow architecture: continuum limits of gauge theories, the mass dependence of QCD observables, and hadronic matrix elements based on the Feynman–Hellmann approach. In all three cases, it is shown that statistical uncertainties are significantly reduced when machine-learned flows are incorporated as compared with the same calculations performed with uncorrelated ensembles or direct reweighting. Published by the American Physical Society 2024 

pdf/applications-of-flow-models-to-the-generation-of-correlated-14y3dcb25i.pdf

Applications of flow models to the generation of correlated lattice QCD ensembles

The traditional Markov chain Monte Carlo (MCMC) suffers from the problem of critical slowing down. Generative machine learning methods, such as normalizing flows, offer a promising method to speed up MCMC simulations, especially in critical regions of lattice field theory. However, training these models for different parameter values in the critical region is inefficient. In this paper, we address this issue by interpolating or extrapolating the flow model to any parameter value in the critical region. We demonstrate the effectiveness of the proposed method for MCMC sampling in critical regions. 

/pdf/conditional-normalizing-flow-for-markov-chain-monte-carlo-1s94zr1r.pdf

Conditional normalizing flow for Markov chain Monte Carlo sampling in the critical region of lattice field theory

The cost of Monte Carlo sampling of lattice conﬁgurations is very high in the critical region of lattice ﬁeld theory due to the high correlation between the samples. This paper suggests a Conditional Normalizing Flow (C-NF) model for sampling lattice conﬁgurations in the critical region to solve the problem of critical slowing down. We train the C-NF model using samples generated by Hybrid Monte Carlo (HMC) in non-critical regions with low simulation costs. The trained C-NF model is employed in the critical region to build a Markov chain of lattice samples with negligible autocorrelation. The C-NF model is used for both interpolation and extrapolation to the critical region of lattice theory. Our proposed method is assessed using the 1+1-dimensional scalar φ 4 theory. This approach enables the construction of lattice ensembles for many parameter values in the critical region, which reduces simulation costs by avoiding the critical slowing down.

Conditional Normalizing flow for Monte Carlo sampling in lattice scalar field theory

Sampling topological quantities in the Monte Carlo simulation of Lattice Gauge Theory becomes challenging as we approach the continuum limit of the theory. In this work, we introduce a Conditional Normalizing Flow (C-NF) model to sample U(1) gauge theory in two dimensions, aiming to mitigate the impact of topological freezing when dealing with smaller values of the U(1) bare coupling. To train the conditional flow model, we utilize samples generated by Hybrid Monte Carlo (HMC) method, ensuring that the autocorrelation in topological quantities remains low. Subsequently, we employ the trained model to interpolate the coupling parameter to values where training was not performed. We thoroughly examine the quality of the model in this region and generate uncorrelated samples, significantly reducing the occurrence of topological freezing. Furthermore, we propose a re-trainable approach that utilizes the model's own samples to enhance the generalization capability of the conditional model. This method enables sampling for coupling values that are far beyond the initial training region, expanding the applicability of the model.

Sampling U(1) gauge theory using a re-trainable conditional flow-based model

pdf/multilevel-generative-samplers-for-investigating-critical-tg4m1ph1g4yq.pdf

Multilevel Generative Samplers for Investigating Critical Phenomena

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Papers

Conditional normalizing flow for Markov chain Monte Carlo sampling in the critical region of lattice field theory

Conditional Normalizing flow for Monte Carlo sampling in lattice scalar field theory

Sampling U(1) gauge theory using a re-trainable conditional flow-based model

Multilevel Generative Samplers for Investigating Critical Phenomena