Abstract
The paper is devoted to the new approach to systems of quasilinear conservation laws which leads to the alternative view of weak solution and to the possibility of developing a new type calculation algorithms for such systems on the basis of neural networks technology. The approach under consideration is the further development of variational point of view to systems of conservation laws that was earlier described by the author. In this paper the multi-dimensional setting is considered but main results are shown in one- and two-dimensional cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
C. M. Dafermos, Conservation Laws in Continuum Physics, Vol. 325 of Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, 2016).
J. S. Hesthaven, Numerical Methods for Conservation Laws: From Analysis to Algorithms (SIAM, Philadelphia, 2018).
D. Tarkhov and A. Vasilyev, Semi-Empirical Neural Network Modeling and Digital Twins Development (Academic, Elsevier, 2020).
S. N. Kruzhkov, ‘‘First order quasilinear equations in several independent variables,’’ Mat. USSR Sb. 10, 217–243 (1970).
A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem (Oxford Univ. Press, New York, 2000).
P. D. Lax, Hyperbolic Partial Differential Equations, Vol. 14 of Courant Lecture Notes in Math. (AMS, Providence, RI, 2006).
Yu. G. Rykov, ‘‘On the variational approach to system of quasilinear conservation laws,’’ Proc. Steklov Inst. Math. 301, 213–227 (2018).
Yu. G. Rykov, ‘‘Extremal properties of the functionals connected with the systems of conservation laws,’’ Math. Montisnigri 46, 21–30 (2019).
C. Michoski, M. Milosavljevic, T. Oliver, and D. R. Hatch, ‘‘Solving differential equations using deep neural networks,’’ Neurocomputing 399, 193–212 (2020).
H. Li, Y. Li, and S. Li, ‘‘Dual neural network method for solving multiple definite integrals,’’ Neural Comput. 31, 208–232 (2019).
S. Changdar and S. Bhattacharjee, ‘‘Solution of definite integrals using functional link artificial neural network,’’ arXiv: 1904.09656v1 (2019).
A. I. Aptekarev and Yu. G. Rykov, ‘‘Variational principle for multidimensional conservation laws and pressureless media,’’ Russ. Math. Surv. 74, 1117–1119 (2019).
Funding
This work was supported by Russian Science Foundation, project no. 19-71-30004.
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by A. I. Aptekarev)
Rights and permissions
About this article
Cite this article
Rykov, Y.G. On the Systems of Conservation Laws and on a New Way To Construct for them Neural Networks Algorithms. Lobachevskii J Math 42, 2645–2653 (2021). https://doi.org/10.1134/S1995080221110184
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080221110184