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.
翻译:我们开发了一个深心神经网络, 重建多边形域的形状, 以其第100个 Laplacian egenvalue 。 有了一个编码器解码器结构, 网络将输入光谱映射到潜在空间, 然后预测方格中域的离散图像。 我们用随机生成的五边形测试了这个网络。 预测的准确性很高, 预测也符合 Laplacian 缩放规则 。 网络恢复了网格对称之外的连续旋转自由度。 缩放变异的潜在变量的变化表明它们与试验多边形的Weyl的参数( 区域、 周边 和 角度的某种函数) 密切相关 。