mlsanityData to 14 Oct 2026

Exponential Convergence of the Deep Neural Network Approximation for Analytic Functions

arXiv 1807.00297 ↗First posted 1 Jul 2018cs.LG stat.MLcited by 38

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Abstract (arXiv)

We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.

Key facts. Cited by 38 papers in the map (data through 14 Oct 2026). First posted on arXiv 1 Jul 2018.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 12; 2022: 10; 2023: 6; 2024: 4; 2025: 1; 2026: 5.

38 papers.

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  1. 2021Optimal Approximation Rate of ReLU Networks in terms of Width and Depthcited by 96
  2. 2021On the approximation of functions by tanh neural networkscited by 81
  3. 2021Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neuronscited by 41
  4. 2022Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuitiescited by 30
  5. 2021Reproducing Activation Function for Deep Learningcited by 19
  6. 2021Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximationcited by 16
  7. 2021Convergence of Deep ReLU Networkscited by 15
  8. 2021Neural Network Approximation of Refinable Functionscited by 15
  9. 2021Convergence of Deep Convolutional Neural Networkscited by 13
  10. 2023On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theoremcited by 13
  11. 2021ReLU Deep Neural Networks from the Hierarchical Basis Perspectivecited by 12
  12. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
26 more
  1. 2022Structural Causal 3D Reconstructioncited by 12
  2. 2024Random Projection Neural Networks of Best Approximation: Convergence theory and practical applicationscited by 11
  3. 2021Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on H"older Classcited by 7
  4. 2022Embedding Principle in Depth for the Loss Landscape Analysis of Deep Neural Networkscited by 7
  5. 2022Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditionscited by 7
  6. 2024Learning smooth functions in high dimensions: from sparse polynomials to deep neural networkscited by 7
  7. 2023Better NTK Conditioning: A Free Lunch from (ReLU) Nonlinear Activation in Wide Neural Networkscited by 6
  8. 2023Uniform Convergence of Deep Neural Networks with Lipschitz Continuous Activation Functions and Variable Widthscited by 5
  9. 2022Theoretical guarantees for the advantage of GNNs over NNs in generalizing bandlimited functions on Euclidean cubescited by 4
  10. 2023What Makes Data Suitable for a Locally Connected Neural Network? A Necessary and Sufficient Condition Based on Quantum Entanglementcited by 4
  11. 2023Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Trainingcited by 4
  12. 2025One model to solve them all: 2BSDE families via neural operatorscited by 4
  13. 2021Analytic function approximation by path norm regularized deep networkscited by 3
  14. 2023Transferability of Graph Neural Networks using Graphon and Sampling Theoriescited by 3
  15. 2022Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activationcited by 2
  16. 2024Optimal Neural Network Approximation for High-Dimensional Continuous Functionscited by 2
  17. 2022Convergence Analysis of Deep Residual Networkscited by 1
  18. 2022Parameter Convex Neural Networkscited by 1
  19. 2024N-Adaptive Ritz Method: A Neural Network Enriched Partition of Unity for Boundary Value Problemscited by 1
  20. 2022A Note on Machine Learning Approach for Computational Imagingcited by 0
  21. 2022Convergence of Deep Neural Networks with General Activation Functions and Poolingcited by 0
  22. 2026Efficient Approximation to Analytic and Lp functions by Height-Augmented ReLU Networkscited by 0
  23. 2026A Composite Activation Function for Learning Stable Binary Representationscited by 0
  24. 2026Approximation of Analytic Functions by ReLU Neural Networks with Adjustable Depth and Widthcited by 0
  25. 2026Spectral Convergence of Random Feature Method in Multiple Dimensionscited by 0
  26. 2026Neural Approximation by Function Composition: Rigidity and Doubly Exponential Convergencecited by 0