# Universal Approximation Using Radial-Basis-Function Networks

@article{Park1991UniversalAU, title={Universal Approximation Using Radial-Basis-Function Networks}, author={Jooyoung Park and Irwin W. Sandberg}, journal={Neural Computation}, year={1991}, volume={3}, pages={246-257} }

There have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables. Some of these studies deal with cases in which the hidden-layer nonlinearity is not a sigmoid. This was motivated by successful applications of feedforward networks with nonsigmoidal hidden-layer units. This paper reports on a related study of radial-basis-function (RBF) networks, and it is proved that RBF networks having one hidden…

## 3,549 Citations

Existence and uniqueness results for neural network approximations

- Computer Science, MathematicsIEEE Trans. Neural Networks
- 1995

Questions of existence and uniqueness of best approximations on a closed interval of the real line under mean-square and uniform approximation error measures are studied and a reparametrization of the class of networks considered in terms of rational functions of a single variable is applied.

Constructive Approximation to Multivariate Function by Decay RBF Neural Network

- Computer ScienceIEEE Transactions on Neural Networks
- 2010

This brief gives a constructive proof for the fact that a decay RBF neural network with n + 1 hidden neurons can interpolate n +1 multivariate samples with zero error and proves that the given decay RBFs can uniformly approximate any continuous multivariate functions with arbitrary precision without training.

Using Radial Basis Function Networks for Function Approximation and Classification

- Computer Science
- 2012

Many aspects associated with the RBF network, such as network structure, universal approimation capability, radial basis functions,RBF network learning, structure optimization, normalized RBF networks, application to dynamic system modeling, and nonlinear complex-valued signal processing, are described.

Approximation of nonlinear systems with radial basis function neural networks

- Computer ScienceIEEE Trans. Neural Networks
- 2001

A technique for approximating a continuous function of n variables with a radial basis function (RBF) neural network is presented, which significantly reduces the network training and evaluation time and the resulting system is bounded-input bounded-output stable.

Neural Networks for Optimal Approximation of Smooth and Analytic Functions

- Mathematics, Computer ScienceNeural Computation
- 1996

We prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives, if the activation…

NEURAL NETWORKS FOR OPTIMAL APPROXIMATION OF SMOOTH

- Mathematics, Computer Science
- 1996

We prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives, if the activation…

Approximation of multivariate functions using ridge polynomial networks

- Computer Science[Proceedings 1992] IJCNN International Joint Conference on Neural Networks
- 1992

A novel class of higher-order feedforward neural networks, called the ridge polynomial network (RPN), is formulated. The networks are shown to uniformly approximate any continuous function of a…

Approximation to a Compact Set of Functions by Feedforward Neural Networks

- Computer Science2007 International Joint Conference on Neural Networks
- 2007

If a family of feedforward neural networks is dense in H, a complete linear metric space of functions, then given a compact set V sub H and an error bound epsiv, one can fix the quantity of the hidden neurons and the weights between the input and hidden layers, such that in order to approximate any function f isin V with accuracy epsv, one only has to further choose suitable weightsbetween the hidden and output layers.

Universal Approximation Using Feedforward Neural Networks: A Survey of Some Existing Methods, and Some New Results

- Computer ScienceNeural Networks
- 1998

The rate of approximation of Gaussian radial basis neural networks in continuous function space

- Computer Science, Mathematics
- 2013

It is proved that the rate of approximation by GRNFNs with nd neurons to any continuous function f defined on a compact subset K ⊂ ℝd can be controlled by ω(f,n−1/2), where ω (f, t) is the modulus of continuity of the function f.

## References

SHOWING 1-10 OF 19 REFERENCES

Approximation by superpositions of a sigmoidal function

- Computer ScienceMath. Control. Signals Syst.
- 1989

In this paper we demonstrate that finite linear combinations of compositions of a fixed, univariate function and a set of affine functionals can uniformly approximate any continuous function ofn real…

Radial Basis Functions, Multi-Variable Functional Interpolation and Adaptive Networks

- Computer Science
- 1988

Abstract : The relationship between 'learning' in adaptive layered networks and the fitting of data with high dimensional surfaces is discussed. This leads naturally to a picture of 'generalization…

Multilayer feedforward networks are universal approximators

- Computer Science, MathematicsNeural Networks
- 1989

Layered Neural Networks with Gaussian Hidden Units as Universal Approximations

- Computer ScienceNeural Computation
- 1990

A neural network with a single layer of hidden units of gaussian type is proved to be a universal approximator for real-valued maps defined on convex, compact sets of Rn.

Pattern classification using neural networks

- Computer ScienceIEEE Communications Magazine
- 1989

The author extends a previous review and focuses on feed-forward neural-net classifiers for static patterns with continuous-valued inputs, examining probabilistic, hyperplane, kernel, and exemplar classifiers.

The Fourier transform.

- PhysicsScientific American
- 1989

The Fourier transform is a function that describes the amplitude and phase of each sinusoid, which corresponds to a specific frequency, which has become a powerful tool in diverse fields of science.

Real And Abstract Analysis

- Economics
- 1965

The first € price and the £ and $ price are net prices, subject to local VAT, and the €(D) includes 7% for Germany, the€(A) includes 10% for Austria.

Radial basis functions for multi-variable interpolation: A

- 1985

Radial basis functions for multi-variable interpolation: A review

- IMA Conference on Algorithms for the Approximation of Functions and Data, RMCS Shrivenham, UK.
- 1985

Gaussian basis functions and approximations for nonlinear systems

- Proceedings of the Nin th Kobe International Symposium on Electronics and Information Sciences
- 1991