KMS Of Academy of mathematics and systems sciences, CAS
| Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units | |
| Li, Bo1,2,3; Tang, Shanshan4; Yu, Haijun1,2,3 | |
| 2020-02-01 | |
| Source Publication | COMMUNICATIONS IN COMPUTATIONAL PHYSICS
 |
| ISSN | 1815-2406 |
| Volume | 27Issue:2Pages: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 epsilon, by our constructive proofs, are in general O(log1/epsilon) 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. |
| Keyword | Deep neural network high dimensional approximation sparse grids rectified linear unit rectified power unit rectified quadratic unit |
| DOI | 10.4208/cicp.OA-2019-0168 |
| Indexed By | SCI |
| Language | 英语 |
| Funding Project | China National Program on Key Basic Research Project[2015CB856003] ; NNSFC[11771439] ; NNSFC[91852116] ; China Science Challenge Project[TZ2018001] |
| WOS Research Area | Physics |
| WOS Subject | Physics, Mathematical |
| WOS ID | WOS:000501534800002 |
| Publisher | GLOBAL SCIENCE PRESS |
| Citation statistics | |
| Document Type | 期刊论文 |
| Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/50326 |
| Collection | 中国科学院数学与系统科学研究院 |
| Corresponding Author | Yu, Haijun |
| Affiliation | 1.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, NCMIS, Beijing 100190, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, LSEC, Beijing 100190, Peoples R China 3.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China 4.China Justice Big Data Inst, Beijing 100043, Peoples R China |
| Recommended Citation GB/T 7714 | Li, Bo,Tang, Shanshan,Yu, Haijun. Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units[J]. COMMUNICATIONS IN COMPUTATIONAL PHYSICS,2020,27(2):379-411. |
| APA | Li, Bo,Tang, Shanshan,&Yu, Haijun.(2020).Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units.COMMUNICATIONS IN COMPUTATIONAL PHYSICS,27(2),379-411. |
| MLA | Li, Bo,et al."Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units".COMMUNICATIONS IN COMPUTATIONAL PHYSICS 27.2(2020):379-411. |
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