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Related Experiment Videos

Simultaneous L(p)-approximation order for neural networks.

Zong-Ben Xu1, Fei-Long Cao

  • 1Faculty of Science, Institute for Information and System Science, Xi'an Jiaotong University, Xi'an, Shaan'xi 710049, People's Republic of China. zbxu@mail.xjtu.edu.cn

Neural Networks : the Official Journal of the International Neural Network Society
|June 7, 2005
PubMed
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This study introduces a novel feedforward neural network (FNN) construction for simultaneous function and derivative approximation. The research provides quantitative accuracy bounds, showing approximation speed depends on network size and function smoothness.

Area of Science:

  • Numerical Analysis
  • Machine Learning
  • Computational Science

Background:

  • Simultaneous approximation of functions and their derivatives is crucial in science and engineering.
  • Existing studies on feedforward neural networks (FNNs) for simultaneous approximation lack quantitative accuracy analysis and topology specification.
  • Previous work focused on density and feasibility, not approximation error bounds in specific metrics.

Purpose of the Study:

  • To construct a class of FNNs capable of simultaneous approximation for smooth multivariate functions and their partial derivatives.
  • To provide a quantitative estimation of the approximation accuracy for these FNNs.
  • To analyze the factors influencing the approximation speed of the proposed FNNs.

Main Methods:

  • Utilized the Bernstein-Durrmeyer operator to construct a novel class of FNNs.

Related Experiment Videos

  • Employed multivariate approximation tools to derive theoretical results.
  • Established quantitative upper bounds on approximation accuracy based on the modulus of smoothness.

    • Main Results:

    • Successfully constructed FNNs that achieve simultaneous approximation of smooth multivariate functions and all their partial derivatives.
    • Derived a quantitative upper bound for the approximation accuracy.
    • Demonstrated that approximation speed is influenced by both the number of hidden units and the function's smoothness.

    Conclusions:

    • The proposed FNNs offer a robust method for simultaneous approximation of functions and their derivatives.
    • The quantitative accuracy bounds provide valuable insights into the performance of these networks.
    • The findings highlight the importance of function smoothness in achieving efficient approximations with FNNs.