Machine learning techniques, in particular the so-called normalizing flows,
are becoming increasingly popular in the context of Monte Carlo simulations as
they can effectively approximate target probability distributions. In the case
of lattice field theories (LFT) the target distribution is given by the
exponential of the action. The common loss function's gradient estimator based
on the "reparametrization trick" requires the calculation of the derivative of
the action with respect to the fields. This can present a significant
computational cost for complicated, non-local actions like e.g. fermionic
action in QCD. In this contribution, we propose an estimator for normalizing
flows based on the REINFORCE algorithm that avoids this issue. We apply it to
two dimensional Schwinger model with Wilson fermions at criticality and show
that it is up to ten times faster in terms of the wall-clock time as well as
requiring up to $30\%$ less memory than the reparameterization trick estimator.
It is also more numerically stable allowing for single precision calculations
and the use of half-float tensor cores. We present an in-depth analysis of the
origins of those improvements. We believe that these benefits will appear also
outside the realm of the LFT, in each case where the target probability
distribution is computationally intensive.

机器学习技术特别是所谓的标准化流在蒙特卡洛模拟中变得越来越受欢迎，因为它们可以有效地逼近目标概率分布。在格点场论中，目标分布由作用的指数给出。我们提出了一种基于 REINFORCE 算法的标准化流估计器，避免了相关的计算问题，应用于临界维度的二维 Schwinger 模型，并显示它相比重新参数化技巧估计器的墙钟时间更快，内存需求减少了 30％，数值上更稳定，并允许进行单精度计算和使用半浮点张量核心。我们深入分析了这些改进的原因，这些优点也将出现在目标概率分布计算复杂的其他领域中。

训练计算密集型目标概率分布的正则化流

Training normalizing flows with computationally intensive target  probability distributions

This notebook tutorial demonstrates a method for sampling Boltzmann
distributions of lattice field theories using a class of machine learning
models known as normalizing flows. The ideas and approaches proposed in
arXiv:1904.12072, arXiv:2002.02428, and arXiv:2003.06413 are reviewed and a
concrete implementation of the framework is presented. We apply this framework
to a lattice scalar field theory and to U(1) gauge theory, explicitly encoding
gauge symmetries in the flow-based approach to the latter. This presentation is
intended to be interactive and working with the attached Jupyter notebook is
recommended.

本文介绍一种使用正则化流方法对晶格场论的 Boltzmann 分布进行采样的方法，特别是在 U（1）规范理论中应用流动方法的编码，以产生包含规范对称性的物理量的结果。

晶格场论中的正则流介绍

Introduction to Normalizing Flows for Lattice Field Theory

A Markov chain update scheme using a machine-learned flow-based generative
model is proposed for Monte Carlo sampling in lattice field theories. The
generative model may be optimized (trained) to produce samples from a
distribution approximating the desired Boltzmann distribution determined by the
lattice action of the theory being studied. Training the model systematically
improves autocorrelation times in the Markov chain, even in regions of
parameter space where standard Markov chain Monte Carlo algorithms exhibit
critical slowing down in producing decorrelated updates. Moreover, the model
may be trained without existing samples from the desired distribution. The
algorithm is compared with HMC and local Metropolis sampling for $\phi^4$
theory in two dimensions.

使用基于机器学习流的生成模型的马尔可夫链更新方案，可用于网格场论的蒙特卡罗采样。