A neural network can hear the shape of a drum

We have developed an artificial neural network that reconstructs the shape of a polygonal domain given the first few dozen of its Laplacian (or Schrodinger) eigenvalues. This provides a practical answer to the famous “can one hear the shape of a drum” question posed by M. Kac in 1966. Having an encoder-decoder structure, the network maps input spectra to a latent space and then predicts the discretized image of the domain. We tested this network on randomly generated triangles, quadrangles, and pentagons. The prediction accuracy remains high for all of these shapes. All predictions scale according to the inputs under the Laplacian scaling rule. The network also recovers the continuous rotational degree of freedom beyond the symmetry of the lattice. The variation of the latent variables under the scaling transformation shows they are strongly correlated with the parameters of the domain (area, perimeter, and a certain function of the angles) from the Weyl expansion.