Shape prediction on the basis of spectrum using neural networks

We have developed a deep neural network that reconstructs the shape of a polygonal domain given the first hundred of its Dirichlet 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. Tested on randomly generated pentagons, the predictions prove to be highly accurate for both concave and convex pentagons. Our analysis indicates that the network has discovered fundamental properties of the Laplacian operator, the scaling rule, and the continuous rotational symmetry. Additionally, the latent variables are strongly correlated with Weyl's parameters (area, perimeter, and a certain function of the angles) of the test polygons.