# Some observations on high-dimensional partial differential equations with Barron data

arXiv: https://arxiv.org/abs/2012.01484 · HTML: https://mlsanity.com/paper/2012.01484/

## Key facts

- Cited by 16 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 2 Dec 2020.

## Built on by (most cited first, 16 of 16)

- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Embeddings between Barron spaces with higher order activation functions](https://mlsanity.com/paper/2305.15839/) (2023)
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- [Embedding Inequalities for Barron-type Spaces](https://mlsanity.com/paper/2305.19082/) (2023)
- [Statistical Spatially Inhomogeneous Diffusion Inference](https://mlsanity.com/paper/2312.05793/) (2023)
- [Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspective](https://mlsanity.com/paper/2412.07728/) (2024)
- [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)
- [A Regularity Theory for Static Schr"odinger Equations on ℝd in Spectral Barron Spaces](https://mlsanity.com/paper/2201.10072/) (2022)
- [A Dual Certificate Approach to Sparsity in Infinite-Width Shallow Neural Networks](https://mlsanity.com/paper/2603.17785/) (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)
- [Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation](https://mlsanity.com/paper/2607.27781/) (2026)
- [Approximation Rates for Metaplectic Neural Networks](https://mlsanity.com/paper/2608.08872/) (2026)

## Abstract (arXiv)

We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Consequently, these solutions can be represented efficiently using artificial neural networks, even in high dimension. Conversely, we present examples in which the solution fails to lie in the function space associated to a neural network under consideration.