A novel method for solving the inverse spectral problem with incomplete data

This paper is concerned with an inverse spectral problem of the Dirichlet boundary in a bounded region. We develop a novel method for determining the unknown objects by using a data-driven deep neural network with convolutional and residual layers. The key ingredient of the approach is to extract features from input data, while fully preserving the original features and preventing network degradation. The network parameters are updated based on the reciprocal of the error calculated by the smooth L₁ function. The incomplete eigenvalues are used to achieve the high-precision inversion of the bounded regions. Numerical experiments are provided to demonstrate the effectiveness of our method in solving the inverse spectral problem in both two-dimensional and three-dimensional cases.