神经网络能听到鼓的形状吗？
Can A Neural Network Hear the Shape of A Drum?
摘要：
我们开发了一个深度神经网络，它可以重建多边形域的形状，给出其拉普拉斯特征值的前一百个。具有编码器-解码器结构，网络将输入光谱映射到潜在空间，然后预测正方形网格上的域的离散化图像。我们在随机生成的五边形上测试了这个网络。预测精度高，预测服从Laplacian标度规则。该网络恢复了网格对称性之外的连续旋转自由度。潜变量在尺度变换下的变化表明，它们与测试多边形的Weyl参数（面积、周长和角度的某个函数）密切相关。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
