Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspective
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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.
Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 13 of its 40 references that are in the map. First posted on arXiv 10 Dec 2024.
Builds on
13 of 40 references.
References · 13
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
- 2020Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don'tcited by 94
- 2020Representation formulas and pointwise properties for Barron functionscited by 62
- 2020On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamicscited by 39
- 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
- 2020Complexity Measures for Neural Networks with General Activation Functions Using Path-based Normscited by 19
- 2023Penalising the biases in norm regularisation enforces sparsitycited by 17
- 2020Some observations on high-dimensional partial differential equations with Barron datacited by 16
- 2019Deep Ritz revisitedcited by 10
- 2023Neural Hilbert Ladders: Multi-Layer Neural Networks in Function Spacecited by 5
- 2022Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionalitycited by 4
1 more
- 2023Embedding Inequalities for Barron-type Spacescited by 2
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Citing papers per year: 2026: 2.
2 papers.