Volume 27, Issue 2
Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units

Bo Li, Shanshan Tang & Haijun Yu

Commun. Comput. Phys., 27 (2020), pp. 379-411.

Published online: 2019-12

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  • Abstract

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU network for functions in Sobolev space and Korobov space have recently been made by [D. Yarotsky, Neural Network, 94:103-114, 2017] and [H. Montanelli and Q. Du, SIAM J Math. Data Sci., 1:78-92, 2019], etc. In this paper, we show that deep networks with rectified power units (RePU) can give better approximations for smooth functions than deep ReLU networks. Our analysis bases on classical polynomial approximation theory and some efficient algorithms proposed in this paper to convert polynomials into deep RePU networks of optimal size with no approximation error. Comparing to the results on ReLU networks, the sizes of RePU networks required to approximate functions in Sobolev space and Korobov space with an error tolerance ε, by our constructive proofs, are in general $O$($log\frac{1}{ε}$) times smaller than the sizes of corresponding ReLU networks constructed in most of the existing literature. Comparing to the classical results of Mhaskar [Mhaskar, Adv. Comput. Math. 1:61-80, 1993], our constructions use less number of activation functions and numerically more stable, they can be served as good initials of deep RePU networks and further trained to break the limit of linear approximation theory. The functions represented by RePU networks are smooth functions, so they naturally fit in the places where derivatives are involved in the loss function.

  • Keywords

Deep neural network, high dimensional approximation, sparse grids, rectified linear unit, rectified power unit, rectified quadratic unit.

  • AMS Subject Headings

65D15, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

libo1171309228@lsec.cc.ac.cn (Bo Li)

tangshanshan@lsec.cc.ac.cn (Shanshan Tang)

hyu@lsec.cc.ac.cn (Haijun Yu)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-27-379, author = {Li , Bo and Tang , Shanshan and Yu , Haijun }, title = {Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units}, journal = {Communications in Computational Physics}, year = {2019}, volume = {27}, number = {2}, pages = {379--411}, abstract = {

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU network for functions in Sobolev space and Korobov space have recently been made by [D. Yarotsky, Neural Network, 94:103-114, 2017] and [H. Montanelli and Q. Du, SIAM J Math. Data Sci., 1:78-92, 2019], etc. In this paper, we show that deep networks with rectified power units (RePU) can give better approximations for smooth functions than deep ReLU networks. Our analysis bases on classical polynomial approximation theory and some efficient algorithms proposed in this paper to convert polynomials into deep RePU networks of optimal size with no approximation error. Comparing to the results on ReLU networks, the sizes of RePU networks required to approximate functions in Sobolev space and Korobov space with an error tolerance ε, by our constructive proofs, are in general $O$($log\frac{1}{ε}$) times smaller than the sizes of corresponding ReLU networks constructed in most of the existing literature. Comparing to the classical results of Mhaskar [Mhaskar, Adv. Comput. Math. 1:61-80, 1993], our constructions use less number of activation functions and numerically more stable, they can be served as good initials of deep RePU networks and further trained to break the limit of linear approximation theory. The functions represented by RePU networks are smooth functions, so they naturally fit in the places where derivatives are involved in the loss function.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0168}, url = {http://global-sci.org/intro/article_detail/cicp/13451.html} }
TY - JOUR T1 - Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units AU - Li , Bo AU - Tang , Shanshan AU - Yu , Haijun JO - Communications in Computational Physics VL - 2 SP - 379 EP - 411 PY - 2019 DA - 2019/12 SN - 27 DO - http://doi.org/10.4208/cicp.OA-2019-0168 UR - https://global-sci.org/intro/article_detail/cicp/13451.html KW - Deep neural network, high dimensional approximation, sparse grids, rectified linear unit, rectified power unit, rectified quadratic unit. AB -

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU network for functions in Sobolev space and Korobov space have recently been made by [D. Yarotsky, Neural Network, 94:103-114, 2017] and [H. Montanelli and Q. Du, SIAM J Math. Data Sci., 1:78-92, 2019], etc. In this paper, we show that deep networks with rectified power units (RePU) can give better approximations for smooth functions than deep ReLU networks. Our analysis bases on classical polynomial approximation theory and some efficient algorithms proposed in this paper to convert polynomials into deep RePU networks of optimal size with no approximation error. Comparing to the results on ReLU networks, the sizes of RePU networks required to approximate functions in Sobolev space and Korobov space with an error tolerance ε, by our constructive proofs, are in general $O$($log\frac{1}{ε}$) times smaller than the sizes of corresponding ReLU networks constructed in most of the existing literature. Comparing to the classical results of Mhaskar [Mhaskar, Adv. Comput. Math. 1:61-80, 1993], our constructions use less number of activation functions and numerically more stable, they can be served as good initials of deep RePU networks and further trained to break the limit of linear approximation theory. The functions represented by RePU networks are smooth functions, so they naturally fit in the places where derivatives are involved in the loss function.

Bo Li, Shanshan Tang & Haijun Yu. (2019). Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units. Communications in Computational Physics. 27 (2). 379-411. doi:10.4208/cicp.OA-2019-0168
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