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In general, a `!function approximation`! problem asks us to select a `F33f`_`[function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f among a well-defined class that closely matches ("approximates") a target function in a task-specific way.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] The need for function approximations arises in many branches of `F33f`_`[applied mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Applied_mathematics]`_`f, and `F33f`_`[computer science`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Computer_science]`_`f in particular , such as predicting the growth of microbes in `F33f`_`[microbiology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Microbiology]`_`f.`:cite-ref-0-2-0[`F5bf`_`[2`#cite-note-0-2]`_`f] Function approximations are used where theoretical models are unavailable or hard to compute.`:cite-ref-0-2-1[`F5bf`_`[2`#cite-note-0-2]`_`f]

One can distinguish two major classes of function approximation problems:

First, for known target functions `F33f`_`[approximation theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Approximation_theory]`_`f is the branch of `F33f`_`[numerical analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Numerical_analysis]`_`f that investigates how certain known functions (for example, `F33f`_`[special functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Special_function]`_`f) can be approximated by a specific class of functions (for example, `F33f`_`[polynomials`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polynomial]`_`f or `F33f`_`[rational functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_function]`_`f) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f]

Second, the target function, call it `*g`*, may be unknown; instead of an explicit formula, only a set of points of the form (`*x`*, `*g`*(`*x`*)) is provided. Depending on the structure of the `F33f`_`[domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Domain_of_a_function]`_`f and `F33f`_`[codomain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Codomain]`_`f of `*g`*, several techniques for approximating `*g`* may be applicable. For example, if `*g`* is an operation on the `F33f`_`[real numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f, techniques of `F33f`_`[interpolation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Interpolation]`_`f, `F33f`_`[extrapolation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Extrapolation]`_`f, `F33f`_`[regression analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Regression_analysis]`_`f, and `F33f`_`[curve fitting`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Curve_fitting]`_`f can be used. If the `F33f`_`[codomain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Codomain]`_`f (range or target set) of `*g`* is a finite set, one is dealing with a `F33f`_`[classification`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Statistical_classification]`_`f problem instead.`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]

To some extent, the different problems (regression, classification, `F33f`_`[fitness approximation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fitness_approximation]`_`f) have received a unified treatment in `F33f`_`[statistical learning theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Statistical_learning_theory]`_`f, where they are viewed as `F33f`_`[supervised learning`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Supervised_learning]`_`f problems.

>>Contents

• `F0af`_`[References`#references]`_`f

• `F0af`_`[See also`#see-also]`_`f

-─

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citereflakemeyersklarsorrentitakahashi2007`aLakemeyer, Gerhard; Sklar, Elizabeth; Sorrenti, Domenico G.; Takahashi, Tomoichi (2007-09-04). `*RoboCup 2006: Robot Soccer World Cup X`*. Springer. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-74024-7.

`:cite-note-0-2`!2.`! `F0af`_`[↑`#cite-ref-0-2-0]`_`f `:citerefbasheerhajmeer2000`aBasheer, I.A.; Hajmeer, M. (2000). "Artificial neural networks: fundamentals, computing, design, and application" (PDF). `*Journal of Microbiological Methods`*. `!43`! (1): 3–31. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1016/S0167-7012(00)00201-3. `F33f`_`[PMID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=PMID_(identifier)]`_`f 11084225. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 18267806.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `:citerefmhaskarpai2000`aMhaskar, Hrushikesh Narhar; Pai, Devidas V. (2000). `*Fundamentals of Approximation Theory`*. CRC Press. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-8493-0939-7.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citerefchartechartegarc-aherrera2019`aCharte, David; Charte, Francisco; García, Salvador; Herrera, Francisco (2019-04-01). "A snapshot on nonstandard supervised learning problems: taxonomy, relationships, problem transformations and algorithm adaptations". `*Progress in Artificial Intelligence`*. `!8`! (1): 1–14. `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:1811.12044. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/s13748-018-00167-7. `F33f`_`[ISSN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISSN_(identifier)]`_`f 2192-6360. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 53715158.

>>See also

• `F33f`_`[Approximation theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Approximation_theory]`_`f

• `F33f`_`[Fitness approximation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fitness_approximation]`_`f

• `F33f`_`[Kriging`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kriging]`_`f

• `F33f`_`[Least squares (function approximation)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Least_squares_(function_approximation)]`_`f

• `F33f`_`[Radial basis function network`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Radial_basis_function_network]`_`f

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