mlsanityData to 14 Oct 2026

Approximation of Functionals by Neural Network without Curse of Dimensionality

arXiv 2205.14421 ↗First posted 28 May 2022math.NA cs.LG cs.NAcited by 8

Not on the map. Why?

Abstract (arXiv)

In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is O(1/√m) where m is the size of networks, which overcomes the curse of dimensionality. The key idea of the approximation is to define a Barron spectral space of functionals.

Key facts. Cited by 8 papers in the map (data through 14 Oct 2026). Builds on 21 of its 37 references that are in the map. First posted on arXiv 28 May 2022.

Builds on

21 of 37 references.

References · 21

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  3. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  4. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  5. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  6. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  7. 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 331
  8. 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
  9. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
  10. 2016Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with ℓ1 and ℓ0 Controlscited by 84
  11. 2018A Priori Estimates of the Population Risk for Two-layer Neural Networkscited by 76
  12. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 62
9 more
  1. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  2. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  3. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  4. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41
  5. 2019The Barron Space and the Flow-induced Function Spaces for Neural Network Modelscited by 39
  6. 2018Solving for high dimensional committor functions using artificial neural networkscited by 34
  7. 2021MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEscited by 24
  8. 2020Complexity Measures for Neural Networks with General Activation Functions Using Path-based Normscited by 19
  9. 2020Deep Learning with Functional Inputscited by 13

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 3; 2025: 2; 2026: 1.

8 papers.

  1. 2024Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Losscited by 14
  2. 2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Normscited by 6
  3. 2023Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Normscited by 4
  4. 2024Newton Informed Neural Operator for Computing Multiple Solutions of Nonlinear Partials Differential Equationscited by 2
  5. 2025Learn Singularly Perturbed Solutions via Homotopy Dynamicscited by 2
  6. 2024DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Trainingcited by 1
  7. 2026Sparse-Aware Neural Networks for Nonlinear Functionals: Mitigating the Exponential Dependence on Dimensioncited by 1
  8. 2023Energy stable neural network for gradient flow equationscited by 0