Can A Neural Network Hear the Shape of A Drum?

TL;DR

This paper presents a deep neural network that reconstructs polygonal shapes from Laplacian eigenvalues, demonstrating high accuracy and capturing geometric properties beyond grid symmetry.

Contribution

It introduces a novel encoder-decoder neural network that maps spectral data to polygonal shapes, capturing geometric features and scaling behaviors.

Findings

High prediction accuracy on randomly generated pentagons

Network predictions obey Laplacian scaling rule

Latent variables correlate with geometric parameters

Abstract

We have developed a deep neural network that reconstructs the shape of a polygonal domain given the first hundred of its Laplacian eigenvalues. Having an encoder-decoder structure, the network maps input spectra to a latent space and then predicts the discretized image of the domain on a square grid. We tested this network on randomly generated pentagons. The prediction accuracy is high and the predictions obey the Laplacian scaling rule. The network recovers the continuous rotational degree of freedom beyond the symmetry of the grid. The variation of the latent variables under the scaling transformation shows they are strongly correlated with Weyl' s parameters (area, perimeter, and a certain function of the angles) of the test polygons.