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Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its
IEEE Transactions on Neural Networks
|January 1, 1995
Summary
This study explores neural network capabilities, finding Tauber-Wiener functions are effective activation functions. This research demonstrates neural networks can approximate complex functions and dynamical systems.
Area of Science:
- Computational Mathematics
- Artificial Intelligence
- Functional Analysis
Background:
- Neural networks are powerful computational tools.
- Understanding their functional approximation capabilities is crucial.
- Activation functions play a key role in neural network performance.
Purpose of the Study:
- To systematically investigate the capability of neural networks.
- To identify suitable activation functions for neural networks.
- To explore the application of neural networks in approximating complex mathematical objects and systems.
Main Methods:
- Analysis of Tauber-Wiener functions as activation functions in three-layered neural networks.
- Characterization of Tauber-Wiener functions within the space of generalized functions S'(R(1)).
- Demonstration of approximation capabilities for nonlinear functionals and operators on Banach spaces.
- Application to approximating the global output of dynamical systems.
Main Results:
- Tauber-Wiener functions are suitable activation functions for hidden layers in three-layered neural networks.
- A continuous function in S'(R(1)) is a Tauber-Wiener function if and only if it is not a polynomial.
- Neural networks can approximate nonlinear functionals on compact sets of Banach spaces and nonlinear operators.
- Neural computation can approximate the entire output of dynamical systems, enabling system identification.
Conclusions:
- The findings expand the theoretical understanding of neural network expressivity.
- Tauber-Wiener functions offer a promising class of activation functions.
- Neural networks demonstrate significant potential for approximating complex mathematical structures and dynamic systems.
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