论文标题
使用神经网络预测第一次通过渗透形状
Predicting First Passage Percolation Shapes Using Neural Networks
论文作者
论文摘要
许多随机增长模型具有正确缩放的发现站点集合的属性，随着时间的增长而收敛到某些确定性集。这样的结果称为形状定理。通常，对形状知之甚少。对于$ \ mathbb {z}^d $上的第一次段落渗透，我们只知道该形状是凸，紧凑的，并且继承了$ \ mathbb {z}^d $的所有对称。使用模拟数据，我们构建并拟合神经网络，能够从均值，标准偏差和通道时间分布的百分位数中充分预测一组发现的位点的形状。音符的目的是两个倍。主要目的是为研究人员提供一个新的工具，以使\ textit {迅速}从通道时间的分布中获得形状的印象 - 而不必等待一些时间才能运行模拟，这是当今唯一可用的方式。该注释的第二个目的只是将现代机器学习方法引入离散概率的领域，并希望它刺激进一步的研究。
Many random growth models have the property that the set of discovered sites, scaled properly, converges to some deterministic set as time grows. Such results are known as shape theorems. Typically, not much is known about the shapes. For first passage percolation on $\mathbb{Z}^d$ we only know that the shape is convex, compact, and inherits all the symmetries of $\mathbb{Z}^d$. Using simulated data we construct and fit a neural network able to adequately predict the shape of the set of discovered sites from the mean, standard deviation, and percentiles of the distribution of the passage times. The purpose of the note is two-fold. The main purpose is to give researchers a new tool for \textit{quickly} getting an impression of the shape from the distribution of the passage times -- instead of having to wait some time for the simulations to run, as is the only available way today. The second purpose of the note is simply to introduce modern machine learning methods into this area of discrete probability, and a hope that it stimulates further research.
