# Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in L2(ℝd,γd)

arXiv: https://arxiv.org/abs/2111.07080 · HTML: https://mlsanity.com/paper/2111.07080/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 28 references that are in the map.
- First posted on arXiv 13 Nov 2021.

## Builds on (most cited first, 4 of 4)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)

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

- [Optimal Convergence Rates for Neural Operators](https://mlsanity.com/paper/2412.17518/) (2024)
- [Approximation Capabilities of Feedforward Neural Networks with GELU Activations](https://mlsanity.com/paper/2512.21749/) (2025)
- [Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operators](https://mlsanity.com/paper/2603.00971/) (2026)

## Abstract (arXiv)

For artificial deep neural networks, we prove expression rates for analytic functions f:ℝd→ℝ in the norm of L2(ℝd,γd) where d∈ mathbbNcup ∞ . Here γd denotes the Gaussian product probability measure on ℝd. We consider in particular ReLU and ReLUk activations for integer k≥ 2. For d∈mathbbN, we show exponential convergence rates in L2(ℝd,γd). In case d=∞, under suitable smoothness and sparsity assumptions on f:ℝ^mathbbN→ℝ, with γ_∞ denoting an infinite (Gaussian) product measure on ℝ^mathbbN, we prove dimension-independent expression rate bounds in the norm of L2(ℝ^mathbbN,γ_∞). The rates only depend on quantified holomorphy of (an analytic continuation of) the map f to a product of strips in mathbbCd. As an application, we prove expression rate bounds of deep ReLU-NNs for response surfaces of elliptic PDEs with log-Gaussian random field inputs.