USING OF MULTILAYER NEURAL NETWORKS FOR THE SOLVING SYSTEMS OF DIFFERENTIAL EQUATIONS
DOI:
https://doi.org/10.20998/2079-0023.2021.02.13Keywords:
systems of differential equations, artificial neural networks, multilayer neural network, numerical methods, gradient descent method, solution’s error functionAbstract
The article considers the study of methods for numerical solution of systems of differential equations using neural networks. To achieve this goal, the
following interdependent tasks were solved: an overview of industries that need to solve systems of differential equations, as well as implemented a
method of solving systems of differential equations using neural networks. It is shown that different types of systems of differential equations can be
solved by a single method, which requires only the problem of loss function for optimization, which is directly created from differential equations and
does not require solving equations for the highest derivative. The solution of differential equations’ system using a multilayer neural networks is the
functions given in analytical form, which can be differentiated or integrated analytically. In the course of this work, an improved form of construction
of a test solution of systems of differential equations was found, which satisfies the initial conditions for construction, but has less impact on the
solution error at a distance from the initial conditions compared to the form of such solution. The way has also been found to modify the calculation of
the loss function for cases when the solution process stops at the local minimum, which will be caused by the high dependence of the subsequent
values of the functions on the accuracy of finding the previous values. Among the results, it can be noted that the solution of differential equations’
system using artificial neural networks may be more accurate than classical numerical methods for solving differential equations, but usually takes
much longer to achieve similar results on small problems. The main advantage of using neural networks to solve differential equations` system is that
the solution is in analytical form and can be found not only for individual values of parameters of equations, but also for all values of parameters in a
limited range of values.
References
Zadachyn V. M., Konyushenko I. G. Chysel’ni metody: Navchal’nyi posibnyk [Numerical methods]. Kharkiv, KhNEU Publ., 2014. 180 p.
Hayrer E., Wanner G. Reshenie obyknovennyh uravnenij. Nezhestkie sadachi. [Solving ordinary differential equations. Non-rigid tasks]. Moscow, Mir Publ., 1999. 685 p.
Lagaris I. E., Likas A., Fotiadis D. I. Artificial Neural Networks for Solving Ordinary and Partial Differential Equations. IEEE Transactions on Neural Networks. 1998, vol. 9, issue 5, pp. 987– 1000.
Devipriya R., Selvi S. Modelling and Solving Differential Equations using Neural Networks: A Study. International Journal of Computational Intelligence and Informatics. 2020, vol. 10, issue 1, pp. 18–23.
Okereke R. N., Maliki O. S, Oruh B. I. A novel method for solving ordinary differential equations with artificial neural networks. Applied Mathematics. 2021, issue 12, pp. 900–918. DOI: 10.4236/am.2021.1210059.
Tsoulos I. G., Gavrilis D., Glavas E. Solving differential equations with constructed neural networks. Neurocomputing. 2009, vol. 72, issue 10–12, pp. 2385–2391.
Korotkaya L. I. Ispol’sovanie neyronnyh setej pri chislennom reshenii differentsyal’nyh uravneniy [The use of neural networks in the numerical solution of some systems of differential equations]. Eastern European Journal of Enterprise Technologies. 2011, issue 3, no. 4(51), pp. 24–27. ISSN 1729-3774.
Denisyuk O. R. Opredelenie ratsyonal’nyh parametrov chislennogo reshenia system differencial’nyh uravneniy nekotoryh klassov [Determination of rational parameters for the numerical solution of systems of differential equations of some classes]. Visnyk of Kherson National Technical University [Bulletin of the Kherson National Technical University]. Kherson Publ., 2016, no. 3 (58), pp. 208–212. ISSN 2078-4481.
Aarts L. P., Van Der Veer P. Neural network method for solving partial differential equations. Neural Processing Letters. 2001, Vol. 14, no. 3, рp. 261–271.
Sirignano J, Spiliopoulos K. DGM: A deep learning algorithm for solving partial differential equations. Journal of computational physics. 2018, vol. 375, pp. 1339–1364.
Baymani М., Kerayechian A., Effati S. Artificial Neural Networks Approach for Solving Stokes Problem. Applied Mathematics. 2010, no. 01(04), pp. 288–292. DOI:10.4236/am.2010.14037.
Marchenko N. A, Sydorenko G. Yu, Rudenko R. O. Using neural networks to solve the differential equation. Informaciyni systemy ta tehnologii: praci 10th Mighnarodnoii konferencii [Information systems and technologies IST-2021 Proceedings of the 10-th International Scientific and Technical Conference]. Kharkiv, KhNURE Publ., 2021, pp. 125–129.
Downloads
Published
How to Cite
Issue
Section
License
LicenseAuthors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
