Numerical Solution of Inverse Eigenvalue and Inverse Source Problems

The latest years, machine learning has been one of the main directions in the numerical solution of inverse problems, aiming to face the ill-posed nature of these problems. In this talk, we delve into the solution of inverse problems and specifically inverse eigenvalue and inverse source problems, from a machine learning perspective. In the first part, we focus on the inverse Sturm-Liouville eigenvalue problem for sym- metric potentials and the inverse transmission eigenvalue problem for spherically sym- metric refractive indices. We present the main ideas behind supervised machine learning regression and briefly discuss the basic properties of the algorithms we implement, which are k-Nearest Neighbours (kNN), Random Forests (RF) and Neural Networks (MLP). Afterwards, we numerically solve the direct problems using well known methods, in order to produce the spectral data which in turn are used for training the machine learning models. We consider examples of inverse problems and compare the performance of each model to predict the unknown potentials and refractive indices respectively, from a given small set of the lowest eigenvalues. In the second part, we pose the inverse source problem, to identify the number, posi- tions, and strengths of hidden line sources inside a dielectric cylinder. Using classification Neural Networks, we show that we can predict the unknown number of sources with high accuracy. We complete this talk with a discussion on an ongoing work for the inverse source problem to recover the positions and strengths of the sources. Our experiments validate the efficiency of these machine learning models for numerically tackling such inverse problems, providing a proof-of-concept for their applicability in this field. 1. N. Pallikarakis and A. Ntargaras, Application of machine learning regression models to inverse eigenvalue problems, Computers & Mathematics with Applications, 154 (2024). 2. N. Pallikarakis, A. Kalogeropoulos and N. L. Tsitsas, Predicting the number of line sources inside a cylinder using classification neural networks, (2024), (to appear in: 2024 IEEE Int. Symp. Antennas Propag. and ITNC-USNC-URSI Radio Sci. Meet.). 3. N. Pallikarakis, A. Kalogeropoulos and N. L. Tsitsas, Exploring the inverse line- source scattering problem in dielectric cylinders with deep neural networks, (2024), (submitted - under review).