Scaling behavior of cross-entropy loss in the identification of percolation phase transitions.

IF 2.4 3区 物理与天体物理 Q1 Mathematics Physical review. E Pub Date : 2024-11-01 DOI:10.1103/PhysRevE.110.054133
Huiyao Li, Yu Zhao, Bo Yang
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引用次数: 0

Abstract

The cross-entropy loss function is widely used in machine learning to measure the performance of a classification model. Interestingly, our study find that this function has scaling behavior when deep neural networks are used to investigate percolation models. Specifically, we use convolutional neural networks with different pooling methods to study the site percolation on square lattices under two labeling methods (labeling based on spanning cluster and the exact solution of the critical point). Subsequently, graph convolutional neural networks (GCNs) with different pooling methods are utilized to do the same kind of experiment. Finally, the GCN with different pooling methods is used to study the percolation phase transitions on the Erdős-Rényi (ER) networks under labeling based on the critical point. The reliability of the classifiers is detected by the values of the critical point p_{c} and critical exponent ν which are obtained by the scaling behaviors of the percolation probability. The results demonstrate that the scaling exponent of cross-entropy ψ/ν depends on the labeling and pooling methods. Labeling based on critical points, which is equivalent to labeling based on spanning clusters in infinite systems, can be used to investigate the critical behaviors in finite systems. SAGPooling-Mean is an effective pooling method to study the scaling behavior of cross-entropy loss on two-dimensional square lattices and ER networks.

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交叉熵损失函数在机器学习中被广泛用于衡量分类模型的性能。有趣的是,我们的研究发现,当使用深度神经网络研究渗滤模型时,该函数具有缩放行为。具体来说,我们使用不同池化方法的卷积神经网络研究了两种标注方法(基于跨簇的标注和临界点的精确解)下方格上的站点渗滤。随后,采用不同池化方法的图卷积神经网络(GCN)也做了同样的实验。最后,不同池化方法的 GCN 被用来研究基于临界点标记的厄尔多斯-雷尼(ER)网络的渗滤相变。分类器的可靠性通过临界点 p_{c} 和临界指数 ν 的值来检测,而临界点 p_{c} 和临界指数 ν 是通过渗滤概率的缩放行为得到的。结果表明,交叉熵 ψ/ν 的缩放指数取决于标记和集合方法。基于临界点的标注相当于无限系统中基于跨簇的标注,可用于研究有限系统中的临界行为。SAGPooling-Mean 是一种有效的集合方法,可用于研究二维方格网和 ER 网络上交叉熵损失的缩放行为。
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来源期刊
Physical review. E
Physical review. E 物理-物理:流体与等离子体
CiteScore
4.60
自引率
16.70%
发文量
0
审稿时长
3.3 months
期刊介绍: Physical Review E (PRE), broad and interdisciplinary in scope, focuses on collective phenomena of many-body systems, with statistical physics and nonlinear dynamics as the central themes of the journal. Physical Review E publishes recent developments in biological and soft matter physics including granular materials, colloids, complex fluids, liquid crystals, and polymers. The journal covers fluid dynamics and plasma physics and includes sections on computational and interdisciplinary physics, for example, complex networks.
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