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On the approximation by single hidden layer feedforward neural networks with fixed weights.

Namig J Guliyev1, Vugar E Ismailov1

  • 1Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, 9 B. Vahabzadeh str., AZ1141, Baku, Azerbaijan.

Neural Networks : the Official Journal of the International Neural Network Society
|January 5, 2018
PubMed
Summary

Single hidden layer feedforward neural networks (SLFNs) with fixed weights can approximate continuous univariate functions using just two neurons. This study constructs a universal sigmoidal activation function for this purpose.

Keywords:
Activation functionApproximationFeedforward neural networkHidden layerSigmoidal functionWeight

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Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Neural Networks

Background:

  • Single hidden layer feedforward neural networks (SLFNs) with fixed weights are known to possess universal approximation properties for univariate functions.
  • The number of neurons in the hidden layer influences the precision of SLFNs, but no inherent restrictions exist.
  • Approximating complex functions often requires a significant number of neurons or adaptable weights.

Purpose of the Study:

  • To constructively prove that SLFNs with a fixed weight of 1 and only two hidden neurons can approximate any continuous univariate function.
  • To develop a universal sigmoidal activation function with desirable properties for function approximation.
  • To demonstrate the limitations of fixed-weight SLFNs in approximating multivariate functions.

Main Methods:

  • A constructive proof approach was employed to demonstrate the approximation capability.
  • A novel universal sigmoidal activation function was designed and built step-by-step.
  • The constructed function was evaluated for properties including computability, smoothness, and weak monotonicity.
  • Numerical examples were utilized to showcase the practical application of the findings.

Main Results:

  • It is proven that SLFNs with fixed weight 1 and two hidden neurons can approximate any continuous function on a compact subset of the real line.
  • A computable, smooth, and weakly monotonic universal sigmoidal activation function was successfully constructed.
  • The effectiveness of this approach was validated through various numerical examples.
  • It was shown that fixed-weight SLFNs are incapable of approximating all continuous multivariate functions.

Conclusions:

  • SLFNs with minimal fixed architecture (weight 1, two neurons) can achieve universal approximation for univariate continuous functions.
  • The developed sigmoidal activation function offers a practical tool for function approximation within this constrained network architecture.
  • The study highlights the inherent limitations of fixed-weight SLFNs when dealing with multivariate function approximation problems.