Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 18, 08 December 2023
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Approximation and interpolation with neural network
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Abstract
In this paper we show that multilayer feedforward networks with one single hidden layer.and certain types of activation functions can approximate univariant continuous functions defined on a compact set. arbitrarily well. In particular, our results contain some usual activation functions such as sigmoidal functions, RELU functions and threshold functions. Besides, since interpolation problems are highly related to approximation problem, we demonstrate that a wide range of functions have the ability to interpolate and generalize our results to functions which are not polynomial on R. Compared to existing results by numerous work, our methods are more intuitive and less technical. Lastly, the paper discusses the possibility of combining interpolation property and approximating property together, and demonstrates that given any Riemann integrable functions on a compact set in R, with several points on its graph, the finite combination of monotone sigmoidal functions can pass through these points and approximate the given function arbitrarily well with respect to L^1 (dx) (in the sense of Riemann integral) when the number of points getting large.
Keywords
neural networks, approximation, interpolation
References
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4. Leshno,M. Lin,V.Y., Pinkus,A. and Schocken,S. (1993), Multilayer feedforward neural networks with a non-polynomial activation function can approximate any function, Neural Networks 6, 861-867
5. Itô,Y. Saito,K. (1996) Superposition of linearly independent functions and finite mappings by neural networks, Math Scient 21, 27-33
6. Huang, G.B., Babri, H.A. (1998) Upper bounds on the number of hidden neurons in feedforward networks with arbitrary bounded nonlinear activation functions, IEEE Trans. Neural Networks 9, 224-229
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8. Grafakus. L.(2014) Classical Fourier Analysis (3rd ed.) Springer-Verlag, New York
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Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of the 2nd International Conference on Computing Innovation and Applied Physics
- ISBN (Print)
- 978-1-83558-201-5
- ISBN (Online)
- 978-1-83558-202-2
- Published Date
- 08 December 2023
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/18/20230354
- Copyright
- 08 December 2023
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited