# A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problems

arXiv: https://arxiv.org/abs/2105.01228 · HTML: https://mlsanity.com/paper/2105.01228/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 59 references that are in the map.
- First posted on arXiv 4 May 2021.

## Builds on (most cited first, 18 of 18)

- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [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)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory](https://mlsanity.com/paper/2006.15733/) (2020)
- [The Barron Space and the Flow-induced Function Spaces for Neural Network Models](https://mlsanity.com/paper/1906.08039/) (2019)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Solving for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)
- [Neural network approximation and estimation of classifiers with classification boundary in a Barron class](https://mlsanity.com/paper/2011.09363/) (2020)
- [Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2012.08023/) (2020)
- [A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations](https://mlsanity.com/paper/2101.01708/) (2021)

## Built on by (most cited first, 19 of 19)

- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems](https://mlsanity.com/paper/2112.03749/) (2021)
- [Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective](https://mlsanity.com/paper/2210.12101/) (2022)
- [Solving PDEs on Unknown Manifolds with Machine Learning](https://mlsanity.com/paper/2106.06682/) (2021)
- [Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks](https://mlsanity.com/paper/2307.15285/) (2023)
- [Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces](https://mlsanity.com/paper/2312.08461/) (2023)
- [Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms](https://mlsanity.com/paper/2311.04779/) (2023)
- [Embedding Inequalities for Barron-type Spaces](https://mlsanity.com/paper/2305.19082/) (2023)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Method](https://mlsanity.com/paper/2405.11451/) (2024)
- [Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions](https://mlsanity.com/paper/2508.07559/) (2025)
- [Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces](https://mlsanity.com/paper/2602.19381/) (2026)
- [Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method](https://mlsanity.com/paper/2607.25100/) (2026)
- [A Regularity Theory for Static Schr"odinger Equations on ℝd in Spectral Barron Spaces](https://mlsanity.com/paper/2201.10072/) (2022)
- [Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations](https://mlsanity.com/paper/2603.23072/) (2026)
- [Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEs](https://mlsanity.com/paper/2605.08672/) (2026)
- [Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Method](https://mlsanity.com/paper/2607.06369/) (2026)
- [Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks](https://mlsanity.com/paper/2609.19937/) (2026)

## Abstract (arXiv)

This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schr"odinger operator on a d-dimensional hypercube. We prove that the convergence rate of the generalization error is independent of the dimension d, under the a priori assumption that the ground state lies in a spectral Barron space. We verify such assumption by proving a new regularity estimate for the ground state in the spectral Barron space. The later is achieved by a fixed point argument based on the Krein-Rutman theorem.