机器学习加速LaMET下的准parton分布函数预测

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"这篇研究论文探讨了如何利用机器学习算法来预测拟parton分布函数（quasi-parton distribution functions, qPDFs）的矩阵元，这是在大动量有效理论（Large Momentum Effective Theory, LaMET）框架下进行的。LaMET使得通过晶格量子色动力学（Lattice Quantum Chromodynamics, LQCD）直接计算强子结构的Bjorken-x依赖性成为可能，扩展了对实验数据不足的动量区域的理解。在下一代LaMET结构计算中，需要非常小的晶格间距以减少伪像，并且长威尔逊连杆位移会增加计算成本。文章中，研究者使用了梯度增强决策树和线性模型这两种机器学习算法，针对Kaon和ηs的非极化parton分布函数（PDF）、介子分布幅度（DA）以及核子的胶子PDF的矩阵元素进行预测。结果表明，这两种算法都能可靠地预测目标观测量，尽管预测精度和系统误差有所不同。特别是，由于数据间的高度相关性，从较小位移z到较大位移的预测比对动量p的预测更为准确。" 这篇研究的重点在于利用机器学习技术降低LQCD计算的成本。随着LaMET的发展，直接计算强子结构的Bjorken-x依赖性成为可能，但精细晶格间距和长威尔逊连杆位移带来的高计算成本是一个挑战。为了应对这个问题，研究团队测试了两种机器学习算法：梯度增强决策树和线性模型，用于预测qPDFs的矩阵元。这两种算法在处理高维、复杂的数据集时表现出色，能够捕捉数据之间的复杂关系。实验结果显示，两种算法都能有效地预测矩阵元，但它们的性能和误差范围有所差异。特别地，预测与位移z相关的信息比预测动量p更准确，这可能是由于位移z的变化对qPDFs的影响更容易通过学习算法捕获。此外，由于在LQCD计算中，矩阵元往往具有内在的相关性，因此机器学习算法能够利用这种相关性提高预测的准确性。这项工作展示了机器学习在高效预测LQCD计算中的潜力，有助于减少计算需求，提高强子结构研究的效率。未来的研究可能会进一步优化算法，以适应更复杂的物理问题，并可能扩展到其他领域，如核物理和粒子物理学中的其他计算密集型问题。

through the matching [37]

ðx; μ

Þ¼

dyZ

ðx; y; μ; μ

Þϕ

ðy; μÞ

þ O



QCD

;



ð9Þ

according to LaMET. The quasi-DA can be obtained by

computing the following correlators for K

−

and η

,as

presented in the Refs. [16,17]:

2pt

ðz; P; tÞ¼h0j



P·y



y; tÞγ

z−1

x¼0

ðy þ xˆz; tÞ

× ψ



y þ zˆz; tÞ

ð0; 0Þγ

ð0; 0Þj0ið10Þ

where fψ

;ψ

gare fu;sgfor K

−

and fs;sgfor η

, Uð



xþzÞ

is the W ilson line connecting lattice site



x to



x þ zˆz.

We perform a calculation using gauge ensembles with

clover valence fermions on a 48

×144 lattice with 2þ1þ1

flavors (degenerate up and down, strange, and charm

degrees of freedom) of highly improved staggered quarks

(HISQ) [38] generated by the MILC Collaboration [39].

The lattice spacing a ≈ 0.06 fm, and m

sea

¼ 310 MeV.

Hypercubic (HYP) smearing [40] is applied to the configu-

rations. The bare quark masses and clover parameters are

tuned to recover the lowest pion mass of the staggered quarks

in the sea. Correlators are calculated from momentum-

smearing sources [41] using 20 source locations on each of

the 95 configurations (1900 measurements in total).

We make two predictions using the ML algorithm. One

is to predict the correlators at larger link length z

pred

from

the correlators at z

pred

. The other is to predict the

correlators of larger momentum p

pred

from the correlators

of p

pred

To determine what input data to use for these predictions,

we first check the correlations among datasets with differ-

ent momenta, link lengths and time slices. The results are

shown in Fig. 1. Here, we set the target data to be the

2-point quasi-DA correlators at p

pred

¼ 5, z

pred

¼ 4 with

input data p

¼ 4, z

¼ 4 for p prediction and p

¼ 5,

< 4 for z prediction. We select the time slice t

pred

¼ 7 to

check the correlations.

Despite the larger error, larger time slices have a weaker

correlation with the target data. This suggests that we

should use input data close to the time slice of the target

data. On the other hand, we should be able to extend the

range of momentum or links of the input.

In the training process, we tried different parameters for

learning rate in f0.5; 0.2; 0.1; 0.02; 0.01; 0.005; 0.002g and

the number of estimators in f100; 150; 200; 250; 300g. The

corresponding fit variance are plotted in a heat map with

range [0,1], as shown in Fig. 2. Considering the fit quality

for both p predictions and z predictions, we selected

parameters r ¼ 0.1, N

est

¼ 150 as having highest fit quality

in both cases; these will be used for further meson-DA

predictions.

The datasets were evenly distributed into three parts:

training data, bias-correction data, and unlabeled test data.

In practice, we want to minimize the labeled data size

without sacrificing much prediction quality. We varied the

amount of training data and bias-correction data from 300

to 500, while keeping the number of unlabeled test data

¼ 900 fixed, to look for a best trade-off between

reduced data size and prediction quality. The results are

shown in Fig. 3. When correlation is obvious, small number

of training and bias-correction datasets provides precise

estimate that is very close to the true observations for the

unlabeled dataset. When correlation is vague, the prediction

becomes more precise as one increases the size of the

training or the bias-correction datasets. Based on the plot,

we picked N

¼ 400, N

¼ 500 for further estimations.

FIG. 1. Correlations between target η

DA C

2pt

data at z

pred

¼ 4, p

pred

¼ 5, t

pred

¼ 7 with input data at a different link length

(momentum) and time slice for z prediction (left) and p prediction (right). The correlation decays quickly, especially at larger t.

ZHANG, FAN, LI, LIN, and YOON PHYS. REV. D 101, 034516 (2020)

034516-4

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