# Physics-Informed Neural Networks for Quantum Eigenvalue Problems

arXiv: https://arxiv.org/abs/2203.00451 · HTML: https://mlsanity.com/paper/2203.00451/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 17 references that are in the map.
- First posted on arXiv 24 Feb 2022.

## Builds on (most cited first, 9 of 9)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [One-Shot Transfer Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2110.11286/) (2021)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)
- [Solving Differential Equations Using Neural Network Solution Bundles](https://mlsanity.com/paper/2006.14372/) (2020)
- [Neural Networks Trained to Solve Differential Equations Learn General Representations](https://mlsanity.com/paper/1807.00042/) (2018)
- [Unsupervised Neural Networks for Quantum Eigenvalue Problems](https://mlsanity.com/paper/2010.05075/) (2020)
- [Unsupervised Reservoir Computing for Solving Ordinary Differential Equations](https://mlsanity.com/paper/2108.11417/) (2021)
- [Semi-supervised Neural Networks solve an inverse problem for modeling Covid-19 spread](https://mlsanity.com/paper/2010.05074/) (2020)

## Built on by (most cited first, 18 of 18)

- [Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2407.17611/) (2024)
- [Operator SVD with Neural Networks via Nested Low-Rank Approximation](https://mlsanity.com/paper/2402.03655/) (2024)
- [Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problems](https://mlsanity.com/paper/2209.11134/) (2022)
- [Solving engineering eigenvalue problems with neural networks using the Rayleigh quotient](https://mlsanity.com/paper/2506.04375/) (2025)
- [Collocation-based Robust Variational Physics-Informed Neural Networks (CRVPINN)](https://mlsanity.com/paper/2401.02300/) (2024)
- [STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem](https://mlsanity.com/paper/2510.23986/) (2025)
- [First principles physics-informed neural network for quantum wavefunctions and eigenvalue surfaces](https://mlsanity.com/paper/2211.04607/) (2022)
- [Exact and approximate error bounds for physics-informed neural networks](https://mlsanity.com/paper/2411.13848/) (2024)
- [Application of machine learning regression models to inverse eigenvalue problems](https://mlsanity.com/paper/2212.04279/) (2022)
- [Physics-Informed Neural Networks for Discovering Localised Eigenstates in Disordered Media](https://mlsanity.com/paper/2305.06802/) (2023)
- [Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities](https://mlsanity.com/paper/2410.04096/) (2024)
- [MixFunn: A Neural Network for Differential Equations with Improved Generalization and Interpretability](https://mlsanity.com/paper/2503.22528/) (2025)
- [A ZeNN architecture to avoid the Gaussian trap](https://mlsanity.com/paper/2505.20553/) (2025)
- [Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers](https://mlsanity.com/paper/2511.01927/) (2025)
- [A Schr"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control](https://mlsanity.com/paper/2603.23173/) (2026)
- [Python library supporting Discrete Variational Formulations and training solutions with Collocation-based Robust Variational Physics Informed Neural Networks (DVF-CRVPINN)](https://mlsanity.com/paper/2604.15398/) (2026)
- [Physics-Informed Neural Networks for Complex Eigenfrequency Identification and Mode Structure Reconstruction of the Ground-State ITG Branch](https://mlsanity.com/paper/2608.01850/) (2026)
- [Physics-Informed Classical and Quantum Neural Networks for One-Dimensional Schrodinger Eigenvalue Problems](https://mlsanity.com/paper/2609.22189/) (2026)

## Abstract (arXiv)

Eigenvalue problems are critical to several fields of science and engineering. We expand on the method of using unsupervised neural networks for discovering eigenfunctions and eigenvalues for differential eigenvalue problems. The obtained solutions are given in an analytical and differentiable form that identically satisfies the desired boundary conditions. The network optimization is data-free and depends solely on the predictions of the neural network. We introduce two physics-informed loss functions. The first, called ortho-loss, motivates the network to discover pair-wise orthogonal eigenfunctions. The second loss term, called norm-loss, requests the discovery of normalized eigenfunctions and is used to avoid trivial solutions. We find that embedding even or odd symmetries to the neural network architecture further improves the convergence for relevant problems. Lastly, a patience condition can be used to automatically recognize eigenfunction solutions. This proposed unsupervised learning method is used to solve the finite well, multiple finite wells, and hydrogen atom eigenvalue quantum problems.