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Neural networks for functional approximation and system identification

H N Mhaskar1, N Hahm

  • 1Department of Mathematics, California State University, Los Angeles 90032, USA.

Neural Computation
|January 1, 1997
PubMed
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Generalized translation networks approximate nonlinear functionals on Lp and C spaces. These networks achieve near-optimal approximation orders with simple, noniterative training, avoiding complex optimization like backpropagation.

Area of Science:

  • Applied Mathematics
  • Machine Learning
  • Functional Analysis

Background:

  • Approximating continuous functionals is crucial in applied mathematics and machine learning.
  • Existing methods often require complex iterative training processes.
  • Nonlinear functionals on function spaces like Lp and C present unique approximation challenges.

Purpose of the Study:

  • To develop generalized translation networks for approximating nonlinear, continuous functionals.
  • To establish theoretical lower bounds for the order of approximation.
  • To demonstrate that the proposed networks can achieve near-optimal approximation orders.

Main Methods:

  • Construction of generalized translation networks.
  • Analysis of approximation orders for functionals on Lp ([-1, 1]s) and C ([-1, 1]s).

Related Experiment Videos

  • Derivation of lower bounds on approximation order based on the number of parameters.

    • Main Results:

    • The proposed networks uniformly approximate the target functionals.
    • Theoretical lower bounds on the order of approximation were established.
    • The networks demonstrated near-optimal approximation performance concerning the number of parameters (neurons).
    • A simple, noniterative training method was employed, avoiding backpropagation.

    Conclusions:

    • Generalized translation networks offer an efficient method for approximating nonlinear functionals.
    • The networks provide a practical alternative to traditional iterative training methods.
    • This work advances the theoretical understanding of approximation capabilities in neural networks.