# Sobolev Approximation by Fixed-Size Neural Networks with Arbitrary Accuracy

arXiv: https://arxiv.org/abs/2606.16975 · HTML: https://mlsanity.com/paper/2606.16975/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 46 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 1 of 1)

- [Mathematical theory of deep learning](https://mlsanity.com/paper/2407.18384/) (2024)

## Built on by (most cited first, 1 of 1)

- [Optimal Tradeoffs Between Network Size and Parameter Magnitude in Neural Approximation and Minimax Regression](https://mlsanity.com/paper/2609.25710/) (2026)

## Abstract (arXiv)

In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks. We first show that any function in W2,∞((a,b)d) can be approximated with arbitrary accuracy, measured in the W1,∞-norm, by a fixed-size neural network using the Elementary Universal Activation Function (EUAF). To extend this result to Ws,∞((a,b)d) for s∈mathbbN, we introduce a smooth activation DUAF∞ from the family of Differentiable Universal Activation Functions (DUAFn). We prove that any function in Ws,∞((a,b)d) can be approximated with arbitrary accuracy in the Ws-1,∞-norm by a fixed-size DUAF∞-activated network. We further construct sigmoidal variants widetildeDUAFn and show that, for every 1≤ s≤ n, fixed-size widetildeDUAFn-activated networks still approximate any f∈ Ws,∞((a,b)d) with arbitrary accuracy in the Ws-1,∞-norm. In all these results, the width and depth bounds are computed explicitly, and the proposed activations are elementary.