# On the Representation of Solutions to Elliptic PDEs in Barron Spaces

arXiv: https://arxiv.org/abs/2106.07539 · HTML: https://mlsanity.com/paper/2106.07539/

## Key facts

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

## Builds on (most cited first, 27 of 27)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
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- [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)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [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)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [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)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [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 for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions](https://mlsanity.com/paper/2007.05384/) (2020)
- [Parametric Complexity Bounds for Approximating PDEs with Neural Networks](https://mlsanity.com/paper/2103.02138/) (2021)
- [A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problems](https://mlsanity.com/paper/2105.01228/) (2021)
- [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)
- [Some observations on high-dimensional partial differential equations with Barron data](https://mlsanity.com/paper/2012.01484/) (2020)
- [Sharp Representation Theorems for ReLU Networks with Precise Dependence on Depth](https://mlsanity.com/paper/2006.04048/) (2020)

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- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Deep Equilibrium Based Neural Operators for Steady-State PDEs](https://mlsanity.com/paper/2312.00234/) (2023)
- [Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective](https://mlsanity.com/paper/2210.12101/) (2022)
- [DeepMed: Semiparametric Causal Mediation Analysis with Debiased Deep Learning](https://mlsanity.com/paper/2210.04389/) (2022)
- [Stationary Density Estimation of Ito Diffusions Using Deep Learning](https://mlsanity.com/paper/2109.03992/) (2021)
- [Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces](https://mlsanity.com/paper/2312.08461/) (2023)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms](https://mlsanity.com/paper/2311.04779/) (2023)
- [In-Context Learning of Linear Dynamical Systems with Transformers: Approximation Bounds and Depth-Separation](https://mlsanity.com/paper/2502.08136/) (2025)
- [Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEs](https://mlsanity.com/paper/2404.17868/) (2024)
- [Embedding Inequalities for Barron-type Spaces](https://mlsanity.com/paper/2305.19082/) (2023)
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- [Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspective](https://mlsanity.com/paper/2412.07728/) (2024)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra](https://mlsanity.com/paper/2505.11638/) (2025)
- [Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions](https://mlsanity.com/paper/2508.07559/) (2025)
- [Sobolev Approximation of Deep ReLU Networks in Log-Barron Space](https://mlsanity.com/paper/2601.01295/) (2026)
- [Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces](https://mlsanity.com/paper/2602.19381/) (2026)
- [A Regularity Theory for Static Schr"odinger Equations on ℝd in Spectral Barron Spaces](https://mlsanity.com/paper/2201.10072/) (2022)
- [Escaping Spectral Bias without Backpropagation: Fast Implicit Neural Representations with Extreme Learning Machines](https://mlsanity.com/paper/2602.07603/) (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)

## 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.