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Two-hidden-layer feed-forward networks are universal approximators: A constructive approach
Eduardo Paluzo-Hidalgo1, Rocio Gonzalez-Diaz1, Miguel A Gutiérrez-Naranjo2
1Department of Applied Mathematics I, University of Seville, Seville, Spain.
Summary
Artificial neural networks are universal approximators. This study provides a constructive method for approximating continuous functions on triangulated spaces using two-hidden-layer networks, with approximation accuracy tied to triangulation refinement.
Area of Science:
- Computational Mathematics
- Artificial Intelligence
- Neural Network Theory
Background:
- Artificial neural networks (ANNs) are established as universal function approximators.
- Classical theory confirms the existence of single-hidden-layer feed-forward networks for function approximation on compact sets.
- Challenges remain in constructing practical network architectures for specific function approximation tasks.
Purpose of the Study:
- To present a constructive method for approximating continuous functions on triangulated spaces using ANNs.
- To demonstrate the feasibility of creating a two-hidden-layer feed-forward network with computable weights.
- To establish a relationship between the refinement of a space's triangulation and the network's approximation accuracy.
Main Methods:
- Utilizing a given triangulation of a continuous space.
- Developing a two-hidden-layer feed-forward neural network architecture.
- Computing a concrete set of weights for the network based on the triangulation.
- Analyzing the approximation error as a function of triangulation refinement.
Main Results:
- A constructive algorithm is presented for generating a two-hidden-layer feed-forward network.
- The network is shown to approximate a continuous function defined on a triangulated space.
- The approximation error is directly related to the fineness of the triangulation, offering a quantifiable measure of accuracy.
Conclusions:
- The study provides a practical, constructive approach to ANN function approximation on triangulated spaces.
- The proposed two-hidden-layer network architecture offers a concrete method for implementation.
- The refinement of the triangulation serves as a key parameter for controlling approximation quality.
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