Radial basis function: Difference between revisions

Revision as of 08:09, 28 June 2025

Revision as of 14:14, 28 June 2025

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Radial basis functions are typically used to build up [[function approximation]]s of the form

{{NumBlk||<math display="block">y(\mathbf{x}) = \sum_{i=1}^N w_i \, \varphi(\left\|\mathbf{x} - \mathbf{x}_i\right\|), </math>|{{EquationRef|910}}}}

where the approximating function <math display="inline">y(\mathbf{x})</math> is represented as a sum of <math>N</math> radial basis functions, each associated with a different center <math display="inline">\mathbf{x}_i</math>, and weighted by an appropriate coefficient <math display="inline">w_i.</math> The weights <math display="inline">w_i</math> can be estimated using the matrix methods of [[Weighted least squares|linear least squares]], because the approximating function is ''linear'' in the weights <math display="inline">w_i</math>.

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The sum

{{NumBlk||<math display="block">y(\mathbf{x}) = \sum_{i=1}^N w_i \, \varphi(\left\|\mathbf{x} - \mathbf{x}_i\right\|), </math>|{{EquationRef|1011}}}}

can also be interpreted as a rather simple single-layer type of [[artificial neural network]] called a [[radial basis function network]], with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any [[continuous function]] on a [[Compact space|compact]] interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number <math display="inline">N</math> of radial basis functions is used.

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{{main|Kansa method}}

Radial basis functions are used to approximate functions and so can be used to discretize and numerically solve Partial Differential Equations (PDEs). This was first done in 1990 by E. J. Kansa who developed the first RBF based numerical method. It is called the [[Kansa method]] and was used to solve the elliptic [[Poisson's equation|Poisson equation]] and the linear [[advection-diffusion equation]]. The function values at points <math>\mathbf{x}</math> in the ___domain are approximated by the linear combination of RBFs:

{{NumBlk||<math display="block">u(\mathbf{x}) = \sum_{i=1}^N \lambda_i \, \varphi(\left\|\mathbf{x} - \mathbf{x}_i\right\|),\quad \mathbf{x}\in\R^d </math>|{{EquationRef|1112}}}}

The derivatives are approximated as such:

{{NumBlk||<math display="block">\frac{\partial^n u(\textbf{x})}{\partial x^n} = \sum_{i=1}^N \lambda_i \, \frac{\partial^n}{\partial x^n}\varphi(\left\|\mathbf{x} - \mathbf{x}_i\right\|),\quad \mathbf{x}\in\R^d </math>|{{EquationRef|1213}}}}

where <math>N</math> are the number of points in the discretized ___domain, <math>d</math> the dimension of the ___domain and <math>\lambda</math> the scalar coefficients that are unchanged by the differential operator.<ref>{{Cite journal | last=Kansa | first=E. J. | date=1990-01-01|title=Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations|journal=Computers & Mathematics with Applications|language=en|volume=19 | issue=8 | pages=147–161|doi=10.1016/0898-1221(90)90271-K|issn=0898-1221|doi-access=free}}</ref>