# Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspective

arXiv: https://arxiv.org/abs/2412.07728 · HTML: https://mlsanity.com/paper/2412.07728/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 40 references that are in the map.
- First posted on arXiv 10 Dec 2024.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Complexity Measures for Neural Networks with General Activation Functions Using Path-based Norms](https://mlsanity.com/paper/2009.06132/) (2020)
- [Penalising the biases in norm regularisation enforces sparsity](https://mlsanity.com/paper/2303.01353/) (2023)
- [Some observations on high-dimensional partial differential equations with Barron data](https://mlsanity.com/paper/2012.01484/) (2020)
- [Deep Ritz revisited](https://mlsanity.com/paper/1912.03937/) (2019)
- [Neural Hilbert Ladders: Multi-Layer Neural Networks in Function Space](https://mlsanity.com/paper/2307.01177/) (2023)
- [Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality](https://mlsanity.com/paper/2209.01173/) (2022)
- [Embedding Inequalities for Barron-type Spaces](https://mlsanity.com/paper/2305.19082/) (2023)

## Built on by (most cited first, 2 of 2)

- [Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method](https://mlsanity.com/paper/2607.25100/) (2026)
- [The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning](https://mlsanity.com/paper/2607.25905/) (2026)

## Abstract (arXiv)

For several classes of neural PDE solvers (Deep Ritz, PINNs, DeepONets), the ability to approximate the solution or solution operator to a partial differential equation (PDE) hinges on the abilitiy of a neural network to approximate the solution in the spatial variables. We analyze the capacity of neural networks to approximate solutions to an elliptic PDE assuming that the boundary condition can be approximated efficiently. Our focus is on the Laplace operator with Dirichlet boundary condition on a half space and on neural networks with a single hidden layer and an activation function that is a power of the popular ReLU activation function.