# Solving engineering eigenvalue problems with neural networks using the Rayleigh quotient

arXiv: https://arxiv.org/abs/2506.04375 · HTML: https://mlsanity.com/paper/2506.04375/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 55 references that are in the map.
- First posted on arXiv 4 Jun 2025.

## Builds on (most cited first, 23 of 23)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Deep Generative Models in Engineering Design: A Review](https://mlsanity.com/paper/2110.10863/) (2021)
- [VarNet: Variational Neural Networks for the Solution of Partial Differential Equations](https://mlsanity.com/paper/1912.07443/) (2019)
- [Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs](https://mlsanity.com/paper/1911.02109/) (2019)
- [Conditional physics informed neural networks](https://mlsanity.com/paper/2104.02741/) (2021)
- [Stress field prediction in fiber-reinforced composite materials using a deep learning approach](https://mlsanity.com/paper/2111.05271/) (2021)
- [A deep learning energy method for hyperelasticity and viscoelasticity](https://mlsanity.com/paper/2201.08690/) (2022)
- [Physics-Informed Neural Networks for Quantum Eigenvalue Problems](https://mlsanity.com/paper/2203.00451/) (2022)
- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [Deep autoencoders for physics-constrained data-driven nonlinear materials modeling](https://mlsanity.com/paper/2209.04416/) (2022)
- [Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problems](https://mlsanity.com/paper/2209.11134/) (2022)
- [Latent Diffusion Models for Structural Component Design](https://mlsanity.com/paper/2309.11601/) (2023)
- [Densely connected neural networks for nonlinear regression](https://mlsanity.com/paper/2108.00864/) (2021)
- [Physics-informed neural network for modeling dynamic linear elasticity](https://mlsanity.com/paper/2312.15175/) (2023)

## Built on by (most cited first, 8 of 8)

- [Nonlinear discretizations and Newton's method: characterizing stationary points of regression objectives](https://mlsanity.com/paper/2510.11987/) (2025)
- [Visualizing the loss landscapes of physics-informed neural networks](https://mlsanity.com/paper/2602.05849/) (2026)
- [Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimization](https://mlsanity.com/paper/2602.13513/) (2026)
- [Fast PINN Eigensolvers via Biconvex Reformulation](https://mlsanity.com/paper/2511.00792/) (2025)
- [Learning Eigenstructures of Unstructured Data Manifolds](https://mlsanity.com/paper/2512.01103/) (2025)
- [Smoothness Errors in Dynamics Models and How to Avoid Them](https://mlsanity.com/paper/2602.05352/) (2026)
- [From inverse problems to neural operators: prediction, mechanism, and generalization of data-driven models](https://mlsanity.com/paper/2606.08956/) (2026)
- [Neural networks for spectral optimization](https://mlsanity.com/paper/2609.36047/) (2026)

## Abstract (arXiv)

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.