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This paper introduces deep super ReLU networks (DSRNs) as a method for approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$ for $m\in\mathbb{N}$ with $m\ge 2$ and $1\le p\le +\infty$.
C. De Boor, C. De Boor, A practical guide to splines, Vol. 27, springer New York, 1978
1978
Earlier work this paper cites.
Y. Abu-Mostafa, The Vapnik-Chervonenkis dimension: Information versus complexity in learning, Neural Computation 1 (3) (1989) 312–317
1989
Earlier work this paper cites.
R. A. DeVore, R. Howard, C. Micchelli, Optimal nonlinear approximation, Manuscripta mathematica 63 (4) (1989) 469–478
1989
Earlier work this paper cites.
D. Pollard, Empirical processes: theory and applications, Ims, 1990
1990
Earlier work this paper cites.
P. Werbos, Approximate dynamic programming for real-time control and neural modeling, Handbook of intelligent control (1992)
1992
Earlier work this paper cites.
R. A. DeVore, G. G. Lorentz, Constructive approximation, Vol. 303, Springer Science & Business Media, 1993
1993
Earlier work this paper cites.
H. Mhaskar, Neural networks for optimal approximation of smooth and analytic functions, Neural computation 8 (1) (1996) 164–177
1996
Earlier work this paper cites.
I. Lagaris, A. Likas, D. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Transactions on Neural Networks 9 (5) (1998) 987–1000
1998
Earlier work this paper cites.
M. Anthony, P. Bartlett, et al., Neural network learning: Theoretical foundations, Vol. 9, cambridge university press Cambridge, 1999
1999
Earlier work this paper cites.
L. Györfi, M. Kohler, A. Krzyzak, H. Walk, et al., A distribution-free theory of nonparametric regression, Vol. 1, Springer, 2002
2002
Earlier work this paper cites.
P. L. Bartlett, O. Bousquet, S. Mendelson, Local rademacher complexities, The Annals of Statistics 33 (4) (2005) 1497–1537
2005
Earlier work this paper cites.
S. Brenner, L. Scott, L. Scott, The mathematical theory of finite element methods, Vol. 3, Springer, 2008
2008
Earlier work this paper cites.
X. Glorot, A. Bordes, Y. Bengio, Deep sparse rectifier neural networks, in: Proceedings of the fourteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, 2011, pp. 315–323
2011
Earlier work this paper cites.
K. He, X. Zhang, S. Ren, J. Sun, Delving deep into rectifiers: Surpassing human-level performance on imagenet classification, in: Proceedings of the IEEE international conference on computer vision, 2015, pp. 1026–1034
2015
Earlier work this paper cites.
A. Krizhevsky, I. Sutskever, G. Hinton, Imagenet classification with deep convolutional neural networks, Communications of the ACM 60 (6) (2017) 84–90
2017
Earlier work this paper cites.
W. Czarnecki, S. Osindero, M. Jaderberg, G. Swirszcz, R. Pascanu, Sobolev training for neural networks, Advances in neural information processing systems 30 (2017)
2017
Earlier work this paper cites.
W. E, J. Han, A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics 5 (4) (2017) 349–380
2017
Earlier work this paper cites.
D. Yarotsky, Error bounds for approximations with deep ReLU networks, Neural Networks 94 (2017) 103–114
2017
Earlier work this paper cites.
2018
Earlier work this paper cites.
W. E, B. Yu, The Deep Ritz Method: A deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics 6 (1) (2018)
2018
Earlier work this paper cites.
J. M. Klusowski, A. R. Barron, Approximation by combinations of relu and squared relu ridge functions with ℓ 1 \ell^{1} and ℓ 0 \ell^{0} controls, IEEE Transactions on Information Theory 64 (12) (2018) 7649–7656
2018
Earlier work this paper cites.
P. Petersen, F. Voigtlaender, Optimal approximation of piecewise smooth functions using deep relu neural networks, Neural Networks 108 (2018) 296–330
2018
Cited alongside, same era.
T. Suzuki, Adaptivity of deep relu network for learning in besov and mixed smooth besov spaces: optimal rate and curse of dimensionality, in: International Conference on Learning Representations, 2018
2018
Cited alongside, same era.
M. Raissi, P. Perdikaris, G. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics 378 (2019) 686–707
2019
Cited alongside, same era.
2019
Cited alongside, same era.
T. Mao, D.-X. Zhou, Approximation of functions from korobov spaces by deep convolutional neural networks, Advances in Computational Mathematics 48 (6) (2022) 84
2022
Later among the works it cites.
S. Mishra, R. Molinaro, Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for pdes, IMA Journal of Numerical Analysis 42 (2) (2022) 981–1022
2022
Later among the works it cites.
L. Evans, Partial differential equations, Vol. 19, American Mathematical Society, 2022
2022
Later among the works it cites.
S. Hon, H. Yang, Simultaneous neural network approximation for smooth functions, Neural Networks 154 (2022) 152–164
2022
Later among the works it cites.
J. W. Siegel, J. Xu, Characterization of the variation spaces corresponding to shallow neural networks, Constructive Approximation 57 (3) (2023) 1109–1132
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2019
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2019
Cited alongside, same era.
P. Bartlett, N. Harvey, C. Liaw, A. Mehrabian, Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks, The Journal of Machine Learning Research 20 (1) (2019) 2285–2301
2019
Cited alongside, same era.
Z. Shen, H. Yang, S. Zhang, Deep network approximation characterized by number of neurons, Communications in Computational Physics 28 (5) (2020)
2020
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2020
Cited alongside, same era.
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2020
Cited alongside, same era.
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2020
Cited alongside, same era.
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2021
Cited alongside, same era.
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2023
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Y. Yang, H. Yang, Y. Xiang, Nearly optimal VC-dimension and pseudo-dimension bounds for deep neural network derivatives, Conference on Neural Information Processing Systems (NeurIPS) (2023)
2023
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2023
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2023
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2024
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2024
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2024
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S. Zhang, J. Lu, H. Zhao, Deep network approximation: Beyond relu to diverse activation functions, Journal of Machine Learning Research 25 (35) (2024) 1–39
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