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

arXiv 2105.01228 ↗First posted 4 May 2021math.NA cs.NA math-phcited by 19

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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.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). Builds on 18 of its 59 references that are in the map. First posted on arXiv 4 May 2021.

Builds on

18 of 59 references.

References · 18

  1. 2014Deep Learning in Neural Networks: An Overviewcited by 765
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  6. 2019Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networkscited by 195
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  8. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
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6 more
  1. 2019The Barron Space and the Flow-induced Function Spaces for Neural Network Modelscited by 39
  2. 2020Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approachcited by 35
  3. 2018Solving for high dimensional committor functions using artificial neural networkscited by 34
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  6. 2021A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equationscited by 18

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 3; 2023: 5; 2024: 1; 2025: 1; 2026: 6.

19 papers.

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  1. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  2. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  3. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  4. 2022Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspectivecited by 15
  5. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
  6. 2023Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networkscited by 11
  7. 2023Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spacescited by 5
  8. 2023Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Normscited by 3
  9. 2023Embedding Inequalities for Barron-type Spacescited by 2
  10. 2023Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Modelscited by 1
  11. 2024Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Methodcited by 1
  12. 2025Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditionscited by 1
7 more
  1. 2026Regularity of Second-Order Elliptic PDEs in Spectral Barron Spacescited by 1
  2. 2026Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Methodcited by 1
  3. 2022A Regularity Theory for Static Schr"odinger Equations on ℝd in Spectral Barron Spacescited by 0
  4. 2026Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equationscited by 0
  5. 2026Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEscited by 0
  6. 2026Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Methodcited by 0
  7. 2026Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networkscited by 0