# Exponential Convergence of the Deep Neural Network Approximation for Analytic Functions

arXiv: https://arxiv.org/abs/1807.00297 · HTML: https://mlsanity.com/paper/1807.00297/

## Key facts

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

## Built on by (most cited first, 30 of 38)

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- [Convergence of Deep Convolutional Neural Networks](https://mlsanity.com/paper/2109.13542/) (2021)
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- [Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on H"older Class](https://mlsanity.com/paper/2103.00542/) (2021)
- [Embedding Principle in Depth for the Loss Landscape Analysis of Deep Neural Networks](https://mlsanity.com/paper/2205.13283/) (2022)
- [Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions](https://mlsanity.com/paper/2208.08635/) (2022)
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- [Better NTK Conditioning: A Free Lunch from (ReLU) Nonlinear Activation in Wide Neural Networks](https://mlsanity.com/paper/2305.08813/) (2023)
- [Uniform Convergence of Deep Neural Networks with Lipschitz Continuous Activation Functions and Variable Widths](https://mlsanity.com/paper/2306.01692/) (2023)
- [Theoretical guarantees for the advantage of GNNs over NNs in generalizing bandlimited functions on Euclidean cubes](https://mlsanity.com/paper/2206.05904/) (2022)
- [What Makes Data Suitable for a Locally Connected Neural Network? A Necessary and Sufficient Condition Based on Quantum Entanglement](https://mlsanity.com/paper/2303.11249/) (2023)
- [Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training](https://mlsanity.com/paper/2311.18022/) (2023)
- [One model to solve them all: 2BSDE families via neural operators](https://mlsanity.com/paper/2511.01125/) (2025)
- [Analytic function approximation by path norm regularized deep networks](https://mlsanity.com/paper/2104.02095/) (2021)
- [Transferability of Graph Neural Networks using Graphon and Sampling Theories](https://mlsanity.com/paper/2307.13206/) (2023)
- [Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activation](https://mlsanity.com/paper/2203.13410/) (2022)
- [Optimal Neural Network Approximation for High-Dimensional Continuous Functions](https://mlsanity.com/paper/2409.02363/) (2024)
- [Convergence Analysis of Deep Residual Networks](https://mlsanity.com/paper/2205.06571/) (2022)
- [Parameter Convex Neural Networks](https://mlsanity.com/paper/2206.05562/) (2022)

## Abstract (arXiv)

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