Application of machine learning regression models to inverse eigenvalue problems

TL;DR

This work tackles inverse eigenvalue problems by leveraging supervised regression models trained on spectra generated from direct problems for two settings: a symmetric Sturm–Liouville system and a spherically symmetric transmission problem. Direct solvers (matslice for Sturm–Liouville and spectral-Galerkin for transmission) produce datasets of eigenvalues which train multi-output regressors (kNN, RF, MLP) to recover the unknown potential or refractive index from the lowest eigenvalues. All models show strong predictive performance in out-of-sample tests, with Random Forests and MLP often achieving the highest accuracy as measured by R^2 and RMSE; feature-importance analysis reveals the leading role of the first (and, in some cases, other early) eigenvalues. The results demonstrate the viability of ML-based approaches for inverse spectral problems and suggest pathways to handle more complex geometries, larger discretizations, and higher-dimensional spectra for practical applications such as non-destructive testing and material characterization.

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 Strum-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.

Figures (11)

• Figure 1: A disc with $L$-layers
• Figure 2: Real transmission eigenvalues of the piecewise constant refractive index $(2.4, 7.4, 8.1, 2.1)$ with $d_1=0.25,\ d_2=0.5$ and $d_3=0.75$, for $m=0,1,2,3,4$.
• Figure 3: A decision tree with depth $2$, corresponding to the training data of subsection \ref{['sec421']}.
• Figure 4: A feed-forward ANN with two hidden layers LN.
• Figure 5: Scaling eigenvalues with Standard Scaler.