Solving engineering eigenvalue problems with neural networks using the Rayleigh quotient

2025-06-04

AI Summary

Continuous eigenvalue problems—such as vibration modes and thermal responses—are long overlooked in physics-informed neural network (PINN) research and challenging to solve via standard neural network discretization due to their inherent infinite-dimensional, spectral nature. Method: We propose an end-to-end framework that represents eigenfunctions with neural networks and tightly integrates Rayleigh quotient variational optimization with Gram–Schmidt orthogonalization—enabling the first deep coupling of the Rayleigh quotient with nonlinear neural discretization. Contribution/Results: The method eliminates reliance on linear-algebraic discretization structures and natively supports parametric, high-dimensional, and irregular-domain modeling. Leveraging automatic differentiation and harmonic spectral basis construction, it robustly computes leading eigenvalues and eigenfunctions across diverse mechanical problems. Experiments validate its effectiveness and generalizability for nonlinear, high-dimensional, and parametric eigenanalysis.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV

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• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Solving engineering eigenvalue problems using neural networks
• Addressing lack of neural network treatments for eigenvalue problems
• Proposing Rayleigh quotient with Gram-Schmidt for eigenanalysis

Innovation

• Neural network discretization of eigenfunctions
• Rayleigh quotient for eigenvalue analysis
• Gram-Schmidt orthogonalization procedure