Predicting First Passage Percolation Shapes Using Neural Networks
This work offers a new computational tool for researchers in discrete probability to quickly estimate shapes, though it is incremental as it applies existing neural network methods to a known problem.
The authors tackled the problem of predicting shapes in first passage percolation, a random growth model, by using a neural network trained on simulated data to predict shapes from passage time statistics, achieving adequate predictions and providing a tool for quick shape estimation.
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.