Exploring Inverse Eigenvalue Problems through Machine Learning

Abstract: 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 numerical solution of inverse eigenvalue problems from a machine learning perspective, focusing on the inverse Sturm--Liouville eigenvalue problem for symmetric potentials and the inverse transmission eigenvalue problem for spherically symmetric refractive indices. Firstly, we formulate these eigenvalue problems and pose the numerical solution of the corresponding direct problems, using well-known numerical methods. Next, we present the main ideas behind the 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 and create the spectral data which in turn are used as training data for 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. Our experiments validate the efficiency of these machine learning models for numerically solving inverse eigenvalue problems, providing a proof-of-concept for their applicability in this field.

[1] N. Pallikarakis and A. Ntargaras, Application of machine leraning regression models to inverse eigenvalue problems, Computers & Mathematics with Applications, 154, 2024.

classical analysis and ODEsspectral theory

Audience: researchers in the topic

Seminars on Inverse Problems Theory and Applications