Predicting First Passage Percolation Shapes Using Neural Networks

TL;DR

This paper introduces a neural network approach to predict the shape of discovered sites in first passage percolation, providing a quick alternative to traditional simulation methods and encouraging machine learning integration in discrete probability.

Contribution

It presents a novel neural network model that predicts percolation shapes from passage time distributions, offering a faster tool for researchers and promoting machine learning in the field.

Findings

Neural network accurately predicts percolation shapes from distribution data.

Method reduces time needed to estimate shapes compared to simulations.

Introduces machine learning as a new approach in discrete probability research.

Abstract

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.