mlsanityData to 7 Oct 2026

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

arXiv 2012.01484 ↗First posted 2 Dec 2020math.AP cs.LGcited by 16

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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.

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

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 2; 2023: 4; 2024: 3; 2025: 1; 2026: 5.

16 papers.

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  1. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 55
  2. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  3. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  4. 2023Embeddings between Barron spaces with higher order activation functionscited by 4
  5. 2023Delay-SDE-net: A deep learning approach for time series modelling with memory and uncertainty estimatescited by 2
  6. 2023Embedding Inequalities for Barron-type Spacescited by 2
  7. 2023Statistical Spatially Inhomogeneous Diffusion Inferencecited by 2
  8. 2024Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspectivecited by 2
  9. 2024Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Methodcited by 1
  10. 2025Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditionscited by 1
  11. 2022A Regularity Theory for Static Schr"odinger Equations on ℝd in Spectral Barron Spacescited by 0
  12. 2026A Dual Certificate Approach to Sparsity in Infinite-Width Shallow Neural Networkscited by 0
4 more
  1. 2026Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEscited by 0
  2. 2026Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Methodcited by 0
  3. 2026Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximationcited by 0
  4. 2026Approximation Rates for Metaplectic Neural Networkscited by 0