Neural Network Approximation for Time Splitting Random Functions
In this article we present the multivariate approximation of time splitting random functions defined on a box or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-...
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| Materyal Türü: | Makale |
| Dil: | English |
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MDPI AG
2023-05-01
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| Seri Bilgileri: | Mathematics |
| Konular: | |
| Online Erişim: | https://www.mdpi.com/2227-7390/11/9/2183 |
| _version_ | 1827742886522781696 |
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| author | George A. Anastassiou Dimitra Kouloumpou |
| author_facet | George A. Anastassiou Dimitra Kouloumpou |
| author_sort | George A. Anastassiou |
| collection | DOAJ |
| description | In this article we present the multivariate approximation of time splitting random functions defined on a box or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>,</mo><mi>N</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>,</mo></mrow></semantics></math></inline-formula> by neural network operators of quasi-interpolation type. We achieve these approximations by obtaining quantitative-type Jackson inequalities engaging the multivariate modulus of continuity of a related random function or its partial high-order derivatives. We use density functions to define our operators. These derive from the logistic and hyperbolic tangent sigmoid activation functions. Our convergences are both point-wise and uniform. The engaged feed-forward neural networks possess one hidden layer. We finish the article with a great variety of applications. |
| first_indexed | 2024-03-11T04:12:40Z |
| format | Article |
| id | doaj.art-f90c97c2a19a4b918c68fa39e2553d52 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:12:40Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f90c97c2a19a4b918c68fa39e2553d522023-11-17T23:21:05ZengMDPI AGMathematics2227-73902023-05-01119218310.3390/math11092183Neural Network Approximation for Time Splitting Random FunctionsGeorge A. Anastassiou0Dimitra Kouloumpou1Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USASection of Mathematics, Hellenic Naval Academy, 18539 Piraeus, GreeceIn this article we present the multivariate approximation of time splitting random functions defined on a box or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>,</mo><mi>N</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi><mo>,</mo></mrow></semantics></math></inline-formula> by neural network operators of quasi-interpolation type. We achieve these approximations by obtaining quantitative-type Jackson inequalities engaging the multivariate modulus of continuity of a related random function or its partial high-order derivatives. We use density functions to define our operators. These derive from the logistic and hyperbolic tangent sigmoid activation functions. Our convergences are both point-wise and uniform. The engaged feed-forward neural networks possess one hidden layer. We finish the article with a great variety of applications.https://www.mdpi.com/2227-7390/11/9/2183logistic and hyperbolic sigmoid functionstime splitting random functionneural network approximationquasi-interpolation operatormultivariate modulus of continuitystochastic inequalities |
| spellingShingle | George A. Anastassiou Dimitra Kouloumpou Neural Network Approximation for Time Splitting Random Functions Mathematics logistic and hyperbolic sigmoid functions time splitting random function neural network approximation quasi-interpolation operator multivariate modulus of continuity stochastic inequalities |
| title | Neural Network Approximation for Time Splitting Random Functions |
| title_full | Neural Network Approximation for Time Splitting Random Functions |
| title_fullStr | Neural Network Approximation for Time Splitting Random Functions |
| title_full_unstemmed | Neural Network Approximation for Time Splitting Random Functions |
| title_short | Neural Network Approximation for Time Splitting Random Functions |
| title_sort | neural network approximation for time splitting random functions |
| topic | logistic and hyperbolic sigmoid functions time splitting random function neural network approximation quasi-interpolation operator multivariate modulus of continuity stochastic inequalities |
| url | https://www.mdpi.com/2227-7390/11/9/2183 |
| work_keys_str_mv | AT georgeaanastassiou neuralnetworkapproximationfortimesplittingrandomfunctions AT dimitrakouloumpou neuralnetworkapproximationfortimesplittingrandomfunctions |