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**Approximation of function and its derivatives using radial basis function networks.**
*(English)*
Zbl 1024.65012

Summary: This paper presents a numerical approach, based on radial basis function networks (RBFNs), for the approximation of a function and its derivatives (scattered data interpolation). The approach proposed here is called the indirect radial basis function network (IRBFN) approximation which is compared with the usual direct approach.

In the direct method (DRBFN) the closed form RBFN approximating function is first obtained from a set of training points and the derivative functions are then calculated directly by differentiating such closed form RBFN.

In the indirect method (IRBFN) the formulation of the problem starts with the decomposition of the derivative of the function into RBFs. The derivative expression is then integrated to yield an expression for the original function, which is then solved via the general linear least squares principle, given an appropriate set of discrete data points. The IRBFN method allows the filtering of noise arisen from the interpolation of the original function from a discrete set of data points and produces a greatly improved approximation of its derivatives.

In both cases the input data consists of a set of unstructured discrete data points (function values), which eliminates the need for a discretisation of the domain into a number of finite elements.

The results obtained are compared with those obtained by the feed forward neural network approach where appropriate and the ”finite element” methods. In all examples considered, the IRBFN approach yields a superior accuracy. For example, all partial derivatives up to second order of the function of three variables \(y=x_1^2+x_1x_2-2x_2^2-x_2x_3+x_3^2\) are approximated with at least an order of magnitude better in the \(L_2\)-norm in comparison with the usual DRBFN approach.

In the direct method (DRBFN) the closed form RBFN approximating function is first obtained from a set of training points and the derivative functions are then calculated directly by differentiating such closed form RBFN.

In the indirect method (IRBFN) the formulation of the problem starts with the decomposition of the derivative of the function into RBFs. The derivative expression is then integrated to yield an expression for the original function, which is then solved via the general linear least squares principle, given an appropriate set of discrete data points. The IRBFN method allows the filtering of noise arisen from the interpolation of the original function from a discrete set of data points and produces a greatly improved approximation of its derivatives.

In both cases the input data consists of a set of unstructured discrete data points (function values), which eliminates the need for a discretisation of the domain into a number of finite elements.

The results obtained are compared with those obtained by the feed forward neural network approach where appropriate and the ”finite element” methods. In all examples considered, the IRBFN approach yields a superior accuracy. For example, all partial derivatives up to second order of the function of three variables \(y=x_1^2+x_1x_2-2x_2^2-x_2x_3+x_3^2\) are approximated with at least an order of magnitude better in the \(L_2\)-norm in comparison with the usual DRBFN approach.

### MSC:

65D05 | Numerical interpolation |

41A05 | Interpolation in approximation theory |

41A63 | Multidimensional problems |

### Keywords:

radial basis function networks; function approximation; derivative approximation; filtering of noise; scattered data interpolation; global approximation; comparison of methods; numerical examples; direct method; indirect method; linear least squares principle### Software:

LINPACK
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\textit{N. Mai-Duy} and \textit{T. Tran-Cong}, Appl. Math. Modelling 27, No. 3, 197--220 (2003; Zbl 1024.65012)

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### References:

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