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Approximations of continuous functionals by neural networks with application to dynamic systems.
IEEE Transactions on Neural Networks
|January 1, 1993
Summary
This study demonstrates that neural networks with a single hidden layer can effectively approximate complex functions in infinite-dimensional spaces. These findings advance the field of neural network representation and approximation theory.
Area of Science:
- Computational Mathematics
- Machine Learning Theory
- Functional Analysis
Background:
- Previous research focused on approximating continuous functions in finite-dimensional spaces using neural networks.
- Limited theoretical understanding existed for neural network capabilities in infinite-dimensional function spaces.
Purpose of the Study:
- To establish strong theoretical results on neural network representation for functions in infinite-dimensional spaces.
- To extend approximation capabilities of neural networks beyond finite-dimensional domains.
- To demonstrate the applicability of these neural network approximations to dynamic systems.
Main Methods:
- Utilizing functional analysis principles to define approximation capabilities.
- Developing theoretical frameworks for neural networks with one hidden layer.
- Applying approximation theorems to function spaces like C[a, b] and L(p)[a, b].
Main Results:
- Proving that neural networks with one hidden layer can arbitrarily approximate functionals on compact sets in infinite-dimensional spaces (C[a, b], L(p)[a, b]).
- Establishing these results under very mild conditions.
- Demonstrating the extension of these approximation capabilities to dynamic system outputs.
Conclusions:
- Neural networks offer powerful representation capabilities for complex functions in infinite-dimensional settings.
- The findings significantly enhance the theoretical foundation of neural network approximation.
- The applicability to dynamic systems highlights practical implications for modeling and prediction.
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