Solving engineering eigenvalue problems with neural networks using the Rayleigh quotient

TL;DR

This paper introduces a neural network-based method using the Rayleigh quotient and Gram-Schmidt orthogonalization to solve complex eigenvalue problems in engineering, demonstrating robustness and versatility across various applications.

Contribution

It presents a novel approach combining neural network discretization with the Rayleigh quotient and orthogonalization for eigenvalue problems, addressing challenges in nonlinearity and high-dimensionality.

Findings

Effective for irregular domains and nonlinear eigenproblems

Applicable to high-dimensional eigenanalysis

Enhances spectral methods for PDE solutions

Abstract

From characterizing the speed of a thermal system's response to computing natural modes of vibration, eigenvalue analysis is ubiquitous in engineering. In spite of this, eigenvalue problems have received relatively little treatment compared to standard forward and inverse problems in the physics-informed machine learning literature. In particular, neural network discretizations of solutions to eigenvalue problems have seen only a handful of studies. Owing to their nonlinearity, neural network discretizations prevent the conversion of the continuous eigenvalue differential equation into a standard discrete eigenvalue problem. In this setting, eigenvalue analysis requires more specialized techniques. Using a neural network discretization of the eigenfunction, we show that a variational form of the eigenvalue problem called the "Rayleigh quotient" in tandem with a Gram-Schmidt orthogonalization procedure is a particularly simple and robust approach to find the eigenvalues and their corresponding eigenfunctions. This method is shown to be useful for finding sets of harmonic functions on irregular domains, parametric and nonlinear eigenproblems, and high-dimensional eigenanalysis. We also discuss the utility of harmonic functions as a spectral basis for approximating solutions to partial differential equations. Through various examples from engineering mechanics, the combination of the Rayleigh quotient objective, Gram-Schmidt procedure, and the neural network discretization of the eigenfunction is shown to offer unique advantages for handling continuous eigenvalue problems.