mlsanityData to 7 Oct 2026

Expressivity of Deep Neural Networks

arXiv 2007.04759 ↗First posted 9 Jul 2020cs.LG math.FA stat.MLcited by 31

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Abstract (arXiv)

In this review paper, we give a comprehensive overview of the large variety of approximation results for neural networks. Approximation rates for classical function spaces as well as benefits of deep neural networks over shallow ones for specifically structured function classes are discussed. While the mainbody of existing results is for general feedforward architectures, we also depict approximation results for convolutional, residual and recurrent neural networks.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). First posted on arXiv 9 Jul 2020.

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Citing papers per year: 2021: 7; 2022: 7; 2023: 8; 2024: 4; 2025: 5.

31 papers.

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  1. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  2. 2021Intriguing Properties of Input-dependent Randomized Smoothingcited by 24
  3. 2024On Provable Length and Compositional Generalizationcited by 22
  4. 2022Understanding the Role of Nonlinearity in Training Dynamics of Contrastive Learningcited by 19
  5. 2021Depth separation beyond radial functionscited by 18
  6. 2023Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirscited by 15
  7. 2021Towards Deepening Graph Neural Networks: A GNTK-based Optimization Perspectivecited by 14
  8. 2022Precision Machine Learningcited by 14
  9. 2023Beyond Distribution Shift: Spurious Features Through the Lens of Training Dynamicscited by 11
  10. 2022Deep neural network expressivity for optimal stopping problemscited by 9
  11. 2023Multilevel CNNs for Parametric PDEscited by 9
  12. 2024Non-convergence to global minimizers for Adam and stochastic gradient descent optimization and constructions of local minimizers in the training of artificial neural networkscited by 9
19 more
  1. 2021Size and Depth Separation in Approximating Benign Functions with Neural Networkscited by 7
  2. 2025Non-convergence to the optimal risk for Adam and stochastic gradient descent optimization in the training of deep neural networkscited by 6
  3. 2022Selected Trends in Artificial Intelligence for Space Applicationscited by 5
  4. 2024DeNOTS: Stable Deep Neural ODEs for Time Seriescited by 5
  5. 2021Weighted Graph-Based Signal Temporal Logic Inference Using Neural Networkscited by 4
  6. 2021Approximation Theory of Tree Tensor Networks: Tensorized Multivariate Functionscited by 3
  7. 2022Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problemscited by 3
  8. 2024Mean flow data assimilation using physics-constrained Graph Neural Networkscited by 3
  9. 2023Universal Approximation Property of Hamiltonian Deep Neural Networkscited by 2
  10. 2025AsyReC: A Multimodal Graph-based Framework for Spatio-Temporal Asymmetric Dyadic Relationship Classificationcited by 2
  11. 2021Machine Learning Applications on Neuroimaging for Diagnosis and Prognosis of Epilepsy: A Reviewcited by 1
  12. 2023From Alexnet to Transformers: Measuring the Non-linearity of Deep Neural Networks with Affine Optimal Transportcited by 1
  13. 2023Statistical learning by sparse deep neural networkscited by 1
  14. 2022NExG: Provable and Guided State Space Exploration of Neural Network Control Systems using Sensitivity Approximationcited by 0
  15. 2023Neural Network Predicts Ion Concentration Profiles under Nanoconfinementcited by 0
  16. 2023Deep neural network approximation of composite functions without the curse of dimensionalitycited by 0
  17. 2025LiLaN: A Linear Latent Network as the Solution Operator for Real-Time Solutions to Stiff Nonlinear Ordinary Differential Equationscited by 0
  18. 2025On the algorithmic construction of deep ReLU networkscited by 0
  19. 2025Error analysis for the deep Kolmogorov methodcited by 0