mlsanityData to 14 Oct 2026

Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensions

arXiv 2312.14276 ↗First posted 21 Dec 2023math.NA cs.LG cs.NAcited by 8

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Abstract (arXiv)

In this study, we establish that deep neural networks employing ReLU and ReLU2 activation functions can effectively represent Lagrange finite element functions of any order on various simplicial meshes in arbitrary dimensions. We introduce two novel formulations for globally expressing the basis functions of Lagrange elements, tailored for both specific and arbitrary meshes. These formulations are based on a geometric decomposition of the elements, incorporating several insightful and essential properties of high-dimensional simplicial meshes, barycentric coordinate functions, and global basis functions of linear elements. This representation theory facilitates a natural approximation result for such deep neural networks. Our findings present the first demonstration of how deep neural networks can systematically generate general continuous piecewise polynomial functions on both specific or arbitrary simplicial meshes.

Key facts. Cited by 8 papers in the map (data through 14 Oct 2026). Builds on 18 of its 45 references that are in the map. First posted on arXiv 21 Dec 2023.

Builds on

18 of 45 references.

References · 18

  1. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  2. 2016Error bounds for approximations with deep ReLU networkscited by 574
  3. 2014On the Number of Linear Regions of Deep Neural Networkscited by 381
  4. 2017The Expressive Power of Neural Networks: A View from the Widthcited by 320
  5. 2016Understanding Deep Neural Networks with Rectified Linear Unitscited by 257
  6. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 166
  7. 2020Deep Network Approximation for Smooth Functionscited by 164
  8. 2019Deep Network Approximation Characterized by Number of Neuronscited by 122
  9. 2015Representation Benefits of Deep Feedforward Networkscited by 86
  10. 2016Deep vs. shallow networks : An approximation theory perspectivecited by 63
  11. 2020Neural Network Approximation: Three Hidden Layers Are Enoughcited by 63
  12. 2021Elementary superexpressive activationscited by 36
6 more
  1. 2023Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivativescited by 30
  2. 2020Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularitiescited by 20
  3. 2021Reproducing Activation Function for Deep Learningcited by 19
  4. 2023On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theoremcited by 13
  5. 2021ReLU Deep Neural Networks from the Hierarchical Basis Perspectivecited by 12
  6. 2018Bounds on the Approximation Power of Feedforward Neural Networkscited by 3

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 1; 2024: 4; 2025: 2; 2026: 1.

8 papers.

  1. 2024KAN: Kolmogorov-Arnold Networkscited by 649
  2. 2023Expressivity and Approximation Properties of Deep Neural Networks with ReLUk Activationcited by 9
  3. 2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Normscited by 6
  4. 2025KAC: Kolmogorov-Arnold Classifier for Continual Learningcited by 4
  5. 2024Exponential Expressivity of ReLUk Neural Networks on Gevrey Classes with Point Singularitiescited by 3
  6. 2024Two-hidden-layer ReLU neural networks and finite elementscited by 0
  7. 2024Kolmogorov-Arnold Neural Networks for High-Entropy Alloys Designcited by 0
  8. 2026Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximationscited by 0