Application of machine learning regression models to inverse eigenvalue problems

Abstract

In this work, we study the numerical solution of inverse eigenvalue problems from a machine learning perspective. Two different problems are considered: the inverse Sturm-Liouville eigenvalue problem for symmetric potentials and the inverse transmission eigenvalue problem for spherically symmetric refractive indices. Firstly, we solve the corresponding direct problems to produce the required eigenvalues datasets in order to train the machine learning algorithms. Next, we consider several 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. The supervised regression models we use are k-Nearest Neighbours, Random Forests and Multi-Layer Perceptron. Our experiments show that these machine learning methods, under appropriate tuning on their parameters, can numerically solve the examined inverse eigenvalue problems.