mlsanityData to 7 Oct 2026

On the Representation of Solutions to Elliptic PDEs in Barron Spaces

arXiv 2106.07539 ↗First posted 14 Jun 2021math.NA cs.LG cs.NAcited by 25

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Abstract (arXiv)

Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of d-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is ε-close with respect to the H1 norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension d of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the H1 norm with a dimension-explicit convergence rate.

Key facts. Cited by 25 papers in the map (data through 7 Oct 2026). Builds on 27 of its 52 references that are in the map. First posted on arXiv 14 Jun 2021.

Builds on

27 of 52 references.

References · 27

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  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
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  5. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
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15 more
  1. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  2. 2019A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problemscited by 95
  3. 2016Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with ℓ1 and ℓ0 Controlscited by 84
  4. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  5. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41
  6. 2019The Barron Space and the Flow-induced Function Spaces for Neural Network Modelscited by 39
  7. 2018Solving for high dimensional committor functions using artificial neural networkscited by 34
  8. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  9. 2020Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditionscited by 25
  10. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  11. 2020Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learningcited by 18
  12. 2021A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equationscited by 18
  13. 2021Parametric Complexity Bounds for Approximating PDEs with Neural Networkscited by 18
  14. 2020Some observations on high-dimensional partial differential equations with Barron datacited by 16
  15. 2020Sharp Representation Theorems for ReLU Networks with Precise Dependence on Depthcited by 13

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 5; 2023: 7; 2024: 3; 2025: 4; 2026: 5.

25 papers.

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  1. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  2. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  3. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  4. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  5. 2023Deep Equilibrium Based Neural Operators for Steady-State PDEscited by 19
  6. 2022Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspectivecited by 15
  7. 2022DeepMed: Semiparametric Causal Mediation Analysis with Debiased Deep Learningcited by 10
  8. 2021Stationary Density Estimation of Ito Diffusions Using Deep Learningcited by 9
  9. 2023Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spacescited by 5
  10. 2025A brief review of the Deep BSDE method for solving high-dimensional partial differential equationscited by 5
  11. 2023Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Normscited by 3
  12. 2024Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEscited by 3
13 more
  1. 2023Embedding Inequalities for Barron-type Spacescited by 2
  2. 2023Statistical Spatially Inhomogeneous Diffusion Inferencecited by 2
  3. 2024Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspectivecited by 2
  4. 2023Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Modelscited by 1
  5. 2025In-Context Learning of Linear Dynamical Systems with Transformers: Approximation Bounds and Depth-Separationcited by 1
  6. 2025Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebracited by 1
  7. 2025Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditionscited by 1
  8. 2026Sobolev Approximation of Deep ReLU Networks in Log-Barron Spacecited by 1
  9. 2026Regularity of Second-Order Elliptic PDEs in Spectral Barron Spacescited by 1
  10. 2022A Regularity Theory for Static Schr"odinger Equations on ℝd in Spectral Barron Spacescited by 0
  11. 2026Escaping Spectral Bias without Backpropagation: Fast Implicit Neural Representations with Extreme Learning Machinescited by 0
  12. 2026Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEscited by 0
  13. 2026Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Methodcited by 0