# Approximation and learning of anisotropic and mixed smooth functions by deep ReLU neural networks

arXiv: https://arxiv.org/abs/2605.31152 · HTML: https://mlsanity.com/paper/2605.31152/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 64 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 26 of 26)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery](https://mlsanity.com/paper/1908.01842/) (2019)
- [Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces](https://mlsanity.com/paper/2211.14400/) (2022)
- [Fast convergence rates of deep neural networks for classification](https://mlsanity.com/paper/1812.03599/) (2018)
- [Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov space](https://mlsanity.com/paper/1910.12799/) (2019)
- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networks](https://mlsanity.com/paper/1903.10047/) (2019)
- [Nonparametric regression using over-parameterized shallow ReLU neural networks](https://mlsanity.com/paper/2306.08321/) (2023)
- [Approximation bounds for norm constrained neural networks with applications to regression and GANs](https://mlsanity.com/paper/2201.09418/) (2022)
- [On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networks](https://mlsanity.com/paper/2409.00901/) (2024)
- [A general approximation lower bound in Lp norm, with applications to feed-forward neural networks](https://mlsanity.com/paper/2206.04360/) (2022)
- [On the rates of convergence for learning with convolutional neural networks](https://mlsanity.com/paper/2403.16459/) (2024)
- [Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces](https://mlsanity.com/paper/2501.11275/) (2025)
- [Learning Tensors in Reproducing Kernel Hilbert Spaces with Multilinear Spectral Penalties](https://mlsanity.com/paper/1310.4977/) (2013)

## Built on by (most cited first, 3 of 3)

- [Approximation of Analytic Functions by ReLU Neural Networks with Adjustable Depth and Width](https://mlsanity.com/paper/2607.10589/) (2026)
- [Shallow neural network approximation in mixed Sobolev spaces](https://mlsanity.com/paper/2609.05263/) (2026)
- [Minimax rates for learning spectral Barron functions by deep ReLU neural networks](https://mlsanity.com/paper/2609.39020/) (2026)

## Abstract (arXiv)

This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions. When the error is measured in Lp([0,1]d) norm and the approximator is a network with width W and depth L, recent works have proven the supper approximation rate O((WL)-2s/d) for Besov space Bsq,r([0,1]d) under the Sobolev embedding condition s/d>1/q-1/p. In order to overcome the curse of dimensionality in this rate, we extent this result to anisotropic and mixed smooth function classes. We establish the approximation rate O((WL)^-2tildes) for anisotropic Besov space B^boldsymbolsq,r([0,1]d) with anisotropic smoothness boldsymbols=(s1,dots,sd) under the embedding condition tildes > 1/q-1/p, where the mean smoothness tildes = (sumi=1d si-1)-1. For mixed smooth Besov space MBsq,r([0,1]d) with mixed smoothness s>1/q-1/p, we show that the approximation rate O((WL)-2s) holds up to logarithmic factors. Using these results, we also derive approximation bounds for the composition of anisotropic Besov functions. As an application, it is shown that deep ReLU neural networks can achieve minimax optimal rates up to logarithmic factors for a wide range of smooth function classes.