Skip to main content

2020 | OriginalPaper | Buchkapitel

Algorithm for Solving Ordinary Differential Equations Using Neural Network Technologies

verfasst von : Irina Bolodurina, Denis Parfenov, Lubov Zabrodina

Erschienen in: Convergent Cognitive Information Technologies

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The paper considers the neural network approach for solving the Cauchy problem for ordinary differential equations of the first order based on the representation of the function as a superposition of elementary functions, the algorithm of solving the problem is proposed. The application of the neural network approach allows obtaining the desired solution in the form of a functional dependence that satisfies smoothness conditions. On the basis of a two-layer perceptron, a model of a neural network solution of the problem and a numerical algorithm realizing the search for a solution are built. We developed a program and algorithmic solution of the Cauchy problem. We analyzed the accuracy of the results and its interrelation with the parameters of the neural networks used. The equivalence of the work of the neural network algorithm and the third-order numerical Runge-Kutta algorithm is shown. In addition, the problem of retraining the neural network algorithm for solving the Cauchy problem for first order ordinary differential equations is posed.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
4.
Zurück zum Zitat Zemskova, Yu.N., Gorbachenko, V.I., Artyukhina, E.V.: The using of trust neighbourhood method to solution partial differential equation on neural networks with radial basis function. Izv. Penz. gos. pedagog. univ. im.i V. G. Belinskogo, no. 18(22), pp. 151–158 (2010). (in Russian) Zemskova, Yu.N., Gorbachenko, V.I., Artyukhina, E.V.: The using of trust neighbourhood method to solution partial differential equation on neural networks with radial basis function. Izv. Penz. gos. pedagog. univ. im.i V. G. Belinskogo, no. 18(22), pp. 151–158 (2010). (in Russian)
6.
Zurück zum Zitat Vasilyev, A.N.: Principles and techniques of neural network modeling. In: Vasiliev, A.N., Tarkhov, D.A. (eds.) SPb.: Nestor-History (2014). (in Russian) Vasilyev, A.N.: Principles and techniques of neural network modeling. In: Vasiliev, A.N., Tarkhov, D.A. (eds.) SPb.: Nestor-History (2014). (in Russian)
8.
Zurück zum Zitat Vasilyev, A.N., Tarkhov, D.A.: Construction of approximate neural network models from heterogeneous data. Matematicheskoe Modelirovaniye [Math. Models Comput. Simul.] 19(12), 43–51 (2007). (in Russian)MATH Vasilyev, A.N., Tarkhov, D.A.: Construction of approximate neural network models from heterogeneous data. Matematicheskoe Modelirovaniye [Math. Models Comput. Simul.] 19(12), 43–51 (2007). (in Russian)MATH
9.
Zurück zum Zitat Vasilyev, A.N., Tarkhov, D.A.: Neural network approaches to solution of boundary problems in multidimensional composite areas. Izvestiya TSURE, no. 9, pp. 80–89 (2004). (in Russian) Vasilyev, A.N., Tarkhov, D.A.: Neural network approaches to solution of boundary problems in multidimensional composite areas. Izvestiya TSURE, no. 9, pp. 80–89 (2004). (in Russian)
10.
Zurück zum Zitat Vasilyev, A.N., Tarkhov, D.A.: Neural network as a new universal approach to the numerical solution of problems of mathematical physics. Neurocomputers, no. 7–8, pp. 111–118 (2004). (in Russian) Vasilyev, A.N., Tarkhov, D.A.: Neural network as a new universal approach to the numerical solution of problems of mathematical physics. Neurocomputers, no. 7–8, pp. 111–118 (2004). (in Russian)
11.
Zurück zum Zitat Kolmogorov, A.N.: On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. Dokl. Akad. Nauk SSSR 114(5), 953–956 (1957). (in Russian) Kolmogorov, A.N.: On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. Dokl. Akad. Nauk SSSR 114(5), 953–956 (1957). (in Russian)
Metadaten
Titel
Algorithm for Solving Ordinary Differential Equations Using Neural Network Technologies
verfasst von
Irina Bolodurina
Denis Parfenov
Lubov Zabrodina
Copyright-Jahr
2020
DOI
https://doi.org/10.1007/978-3-030-37436-5_7