# Approximations with deep neural networks in Sobolev time-space

arXiv: https://arxiv.org/abs/2101.06115 · HTML: https://mlsanity.com/paper/2101.06115/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2020.

## Built on by (most cited first, 13 of 13)

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- [Understanding Deep Neural Function Approximation in Reinforcement Learning via ε-Greedy Exploration](https://mlsanity.com/paper/2209.07376/) (2022)
- [Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regression](https://mlsanity.com/paper/2305.00608/) (2023)
- [Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms](https://mlsanity.com/paper/2508.05141/) (2025)
- [SignReLU neural network and its approximation ability](https://mlsanity.com/paper/2210.10264/) (2022)
- [Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks](https://mlsanity.com/paper/2512.23643/) (2025)
- [Integral representations of shallow neural network with Rectified Power Unit activation function](https://mlsanity.com/paper/2112.11157/) (2021)
- [Can neural operators always be continuously discretized?](https://mlsanity.com/paper/2412.03393/) (2024)
- [Time-Frequency Analysis for Neural Networks](https://mlsanity.com/paper/2512.15992/) (2025)
- [Sampling Complexity of Deep Approximation Spaces](https://mlsanity.com/paper/2312.13379/) (2023)
- [Error analysis for the deep Kolmogorov method](https://mlsanity.com/paper/2508.17167/) (2025)
- [Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation](https://mlsanity.com/paper/2607.27781/) (2026)
- [Approximation Rates for Metaplectic Neural Networks](https://mlsanity.com/paper/2608.08872/) (2026)

## Abstract (arXiv)

Solutions of evolution equation generally lies in certain Bochner-Sobolev spaces, in which the solution may has regularity and integrability properties for the time variable that can be different for the space variables. Therefore, in this paper, we develop a framework shows that deep neural networks can approximate Sobolev-regular functions with respect to Bochner-Sobolev spaces. In our work we use the so-called Rectified Cubic Unit (ReCU) as an activation function in our networks, which allows us to deduce approximation results of the neural networks while avoiding issues caused by the non regularity of the most commonly used Rectivied Linear Unit (ReLU) activation function.