Multi-artificial neural network for inverse eigenvalue problem with the weighted Helmholtz equation

The inverse eigenvalue problem of weighted Helmholtz equations is investigated. The density function is recovered from the observation of the limited spectral data. It is reformulated as an optimization problem and a multi-objective loss function is defined accordingly. A multi-artificial neural network (multi-ANN) algorithm is proposed. The existence and stability of the solution of the optimization problem, and the convergence of the multi-ANN solution towards that of the original optimization problem are proved. The numerical results of one-dimensional and two-dimensional inverse eigenvalue problems of the weighted Helmholtz equation are given. Compared with the traditional finite element method, the robustness and effectiveness of the proposed multi-ANN method are illustrated.