Extrapolation of Duffing Equation Solution by Using RBF Network
Abstract
The article considers the problem of approximation of the solution of the Cauchy problem for an ordinary differential equation of the second order. The approximation scheme is based on the Taylor expansion of the solution with a remainder in the Lagrange form. The residual term is sought in the form of the output of a neural network with radial basis functions (RBF networks). An algorithm for learning (choosing the optimal parameters) of the RBF network is presented. Using the example of solving the Cauchy problem for a second-order nonlinear differential equation – the Duffing oscillator, the influence of various radial-basis functions on the quality of interpolation and extrapolation of the solution to the Duffing equation is investigated.
References
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2. Rai R., Sahu C.K. Driven by Data or Derived through Physics? A Review of Hybrid Physics Guided Machine Learning Techniques with Cyber-Physical System (CPS) Focus. IEEE Access. 2020; 8:71050-71073. (In Eng.) doi: https://doi.org/10.1109/ACCESS.2020.2987324
3. Lagaris I.E., Likas A., Fotiadis D.I. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks. 1998; 9(5):987-1000. (In Eng.) doi: https://doi.org/10.1109/72.712178
4. Kumar M., Yadav N. Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey. Computers and Mathematics with Applications. 2011; 62(10):3796-3811. (In Eng.) doi: https://doi.org/10.1016/j.camwa.2011.09.028
5. Mai-Duy N. Solving high order ordinary differential equations with radial basis function networks. International Journal for Numerical Methods in Engineering. 2005; 6(92):824-852. (In Eng.) doi: https://doi.org/10.1002/nme.1220
6. Budkina E.M., Kuznetsov E.B., Lazovskaya T.V., Leonov S.S., Tarkhov D.A., Vasilyev A.N. Neural Network Technique in Boundary Value Problems for Ordinary Differential Equations. In: Ed. by L. Cheng, Q. Liu, A. Ronzhin. Advances in Neural Networks – ISNN 2016. ISNN 2016. Lecture Notes in Computer Science. Vol. 9719. Springer, Cham; 2016. p. 277-283. (In Eng.) doi: https://doi.org/10.1007/978-3-319-40663-3_32
7. Budkina E.M., Kuznetsov E.B., Lazovskaya T.V., Tarkhov D.A., Shemyakina T.A., Vasilyev A.N. Neural network approach to intricate problems solving for ordinary differential equations. Optical Memory and Neural Networks. 2017; 26(2):96-109. (In Eng.) doi: https://doi.org/10.3103/S1060992X17020011
8. Tarkhov D.A., Vasilyev A.N. Semi-Empirical Neural Network Modeling and Digital Twins Development. Academic Press, Elsevier; 2019. 288 p. (In Eng.) doi: https://doi.org/10.1016/C2017-0-02027-X
9. Zhang M., Drikakis D., Li L., Yan X. Machine-Learning Prediction of Underwater Shock Loading on Structures. Computation. 2019; 7(4):58. (In Eng.) doi: https://doi.org/10.3390/computation7040058
10. Egorchev M.V., Tiumentsev Y.V. Semi-Empirical Continuous Time Neural Network Based Models for Controllable Dynamical Systems. Optical Memory and Neural Networks. 2019; 28(3);192-203. (In Eng.) doi: https://doi.org/10.3103/S1060992X1903010X
11. Raissi M., Perdikaris P., Karniadakis G. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics. 2019; 378:686-707. (In Eng.) doi: https://doi.org/10.1016/j.jcp.2018.10.045
12. Han J., Jentzen A., Weinan E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences. 2018; 115(34):8505-8510. (In Eng.) doi: https://doi.org/10.1073/pnas.1718942115
13. Raissi M. Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations. arXiv:1804.07010. 2018. (In Eng.) doi: https://doi.org/10.48550/arXiv.1804.07010
14. Sirignano J., Spiliopoulos K. DGM: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics. 2018; 375:1339-1364. (In Eng.) doi: https://doi.org/10.1016/j.jcp.2018.08.029
15. Hairer E., Wanner G., Nørsett S.P. Solving Ordinary Differential Equations I: Nonstiff Problem. Springer Series in Computational Mathematics. Vol. 8. Springer Berlin, Heidelberg; 1993. 2nd Edition. 528 p. (In Eng.) doi: https://doi.org/10.1007/978-3-540-78862-1
16. Vasilyev A.N., Kolbin I.S., Reviznikov D.L. Meshfree Computational Algorithms Based on Normalized Radial Basis Functions. In: Cheng L., Liu Q., Ronzhin A. Advances in Neural Networks – ISNN 2016. ISNN 2016. Lecture Notes in Computer Science. Vol. 9719. Springer, Cham; 2016. p. 583-591. (In Eng.) DOI: https://doi.org/10.1007/978-3-319-40663-3_67
17. Gorbachenko V.I., Lazovskaya T.V., Tarkhov D.A., Vasilyev A.N., Zhukov M.V. Neural Network Technique in Some Inverse Problems of Mathematical Physics. In: Cheng L., Liu Q., Ronzhin A. (eds.) Advances in Neural Networks – ISNN 2016. ISNN 2016. Lecture Notes in Computer Science. Vol. 9719. Springer, Cham; 2016. p. 310-316. (In Eng.) doi: https://doi.org/10.1007/978-3-319-40663-3_36
18. Linde Y., Buzo A., Gray R. An Algorithm for Vector Quantizer Design. IEEE Transactions on Communications. 1980; 1(28):84-95. (In Eng.) doi: https://doi.org/10.1109/TCOM.1980.1094577
19. Vasilyev A.N., Gorokhovskaya V.A., Korchagin A.P., Lazovskaya T.V., Tarkhov D.A., Chernukha D.A. Investigation of the Predictive Capabilities of a Data-Driven Multilayer Neuromorphic Model by the Example of the Duffing Oscillator. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2021; 17(3):625-632. (In Russ., abstract in Eng.) doi: https://doi.org/10.25559/SITITO.17.202103.625-632
20. Yang X., Li Z., Dahhou B. Parameter and State Estimation for Uncertain Nonlinear Systems by Adaptive Observer Based on Differential Evolution Algorithm. Applied Sciences. 2020; 10(17):5857. (In Eng.) doi: https://doi.org/10.3390/app10175857
21. Pang G., D'Elia M., Parks M., Karniadakis G.E. nPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications. Journal of Computational Physics. 2020; 422:109760. (In Eng.) doi: https://doi.org/10.1016/j.jcp.2020.109760
22. Sands T. Nonlinear-Adaptive Mathematical System Identification. Computation. 2017; 5(4):47. (In Eng.) doi: https://doi.org/10.3390/computation5040047
23. Zhang X., Ding F. Adaptive parameter estimation for a general dynamical system with unknown states. International Journal of Robust and Nonlinear Control. 2020; 30(4):1351-1372. (In Eng.) doi: https://doi.org/10.1002/rnc.4819
24. Lazovskaya T., Tarkhov D. Multilayer neural network models based on grid methods. IOP Conference Series: Materials Science and Engineering. 2016; 158(1):012061. (In Eng.) doi: https://doi.org/10.1088/1757-899X/158/1/012061
25. Boyarsky S., Lazovskaya T., Tarkhov D. Investigation of the Predictive Capabilities of a Data-Driven Multilayer Model by the Example of the Duffing Oscillator. Proceedings of the 2020 International Multi-Conference on Industrial Engineering and Modern Technologies (FarEastCon). IEEE Computer Society, Vladivostok, Russia; 2020. p. 1-5. (In Eng.) doi: https://doi.org/10.1109/FarEastCon50210.2020.9271195
26. Lazovskaya T., Malykhina G., Tarkhov D. Construction of an Individual Model of the Deflection of a PVC-Specimen Based on a Differential Equation and Measurement Data. Proceedings of the 2020 International Multi-Conference on Industrial Engineering and Modern Technologies (FarEastCon). IEEE Computer Society, Vladivostok, Russia; 2020. p. 1-4. (In Eng.) doi: https://doi.org/10.1109/FarEastCon50210.2020.9271144
Published
2021-12-20
How to Cite
LAZOVSKAYA, Tatyana Valerievna et al.
Extrapolation of Duffing Equation Solution by Using RBF Network.
Modern Information Technologies and IT-Education, [S.l.], v. 17, n. 4, p. 998-1006, dec. 2021.
ISSN 2411-1473. Available at: <http://sitito.cs.msu.ru/index.php/SITITO/article/view/775>. Date accessed: 20 sep. 2025.
doi: https://doi.org/10.25559/SITITO.17.202104.998-1006.
Section
Scientific software in education and science
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