Approximations with deep neural networks in Sobolev time-space
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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.
Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). First posted on arXiv 23 Dec 2020.
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Citing papers per year: 2021: 1; 2022: 3; 2023: 2; 2024: 1; 2025: 4; 2026: 2.
13 papers.
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- 2022Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activationscited by 29
- 2022Understanding Deep Neural Function Approximation in Reinforcement Learning via ε-Greedy Explorationcited by 11
- 2023Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regressioncited by 8
- 2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Normscited by 5
- 2022SignReLU neural network and its approximation abilitycited by 4
- 2025Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networkscited by 3
- 2021Integral representations of shallow neural network with Rectified Power Unit activation functioncited by 2
- 2024Can neural operators always be continuously discretized?cited by 2
- 2025Time-Frequency Analysis for Neural Networkscited by 2
- 2023Sampling Complexity of Deep Approximation Spacescited by 1
- 2025Error analysis for the deep Kolmogorov methodcited by 0
- 2026Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximationcited by 0
1 more
- 2026Approximation Rates for Metaplectic Neural Networkscited by 0
- 2026Approximation Rates for Metaplectic Neural Networks9 Aug 2026
- 2026Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation30 Jul 2026
- 2025Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks29 Dec 2025
- 2025Time-Frequency Analysis for Neural Networks17 Dec 2025
- 2025Error analysis for the deep Kolmogorov method23 Aug 2025
- 2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms7 Aug 2025
- 2024Can neural operators always be continuously discretized?4 Dec 2024
- 2023Sampling Complexity of Deep Approximation Spaces20 Dec 2023
- 2023Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regression1 May 2023
- 2022SignReLU neural network and its approximation ability19 Oct 2022
- 2022Understanding Deep Neural Function Approximation in Reinforcement Learning via ε-Greedy Exploration15 Sep 2022
- 2022Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations20 Jun 2022
- 2022Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activationscited by 29
- 2022Understanding Deep Neural Function Approximation in Reinforcement Learning via ε-Greedy Explorationcited by 11
- 2023Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regressioncited by 8
- 2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Normscited by 5
- 2022SignReLU neural network and its approximation abilitycited by 4
- 2025Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networkscited by 3
- 2021Integral representations of shallow neural network with Rectified Power Unit activation functioncited by 2
- 2024Can neural operators always be continuously discretized?cited by 2
- 2025Time-Frequency Analysis for Neural Networkscited by 2
- 2023Sampling Complexity of Deep Approximation Spacescited by 1
- 2025Error analysis for the deep Kolmogorov methodcited by 0
- 2026Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximationcited by 0
12 most cited of 13 All 13 (JSON)
- 2026Approximation Rates for Metaplectic Neural Networks9 Aug 2026
- 2026Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation30 Jul 2026
- 2025Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks29 Dec 2025
- 2025Time-Frequency Analysis for Neural Networks17 Dec 2025
- 2025Error analysis for the deep Kolmogorov method23 Aug 2025
- 2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms7 Aug 2025
- 2024Can neural operators always be continuously discretized?4 Dec 2024
- 2023Sampling Complexity of Deep Approximation Spaces20 Dec 2023
- 2023Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regression1 May 2023
- 2022SignReLU neural network and its approximation ability19 Oct 2022
- 2022Understanding Deep Neural Function Approximation in Reinforcement Learning via ε-Greedy Exploration15 Sep 2022
- 2022Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations20 Jun 2022
12 newest of 13 All 13 (JSON)