Mathematical modeling of distributed systems by neural networks
Matematičeskoe modelirovanie, Tome 19 (2007) no. 12, pp. 32-42.

Voir la notice de l'article provenant de la source Math-Net.Ru

Neural networks are considered as a new universal approach to the mathematical model construction of systems with distributed parameters. They allow us to find efficiently approximate solutions of initial and boundary value problems for partial differential equations and to take into consideration phenomena and coefficient perturbations.
@article{MM_2007_19_12_a3,
     author = {A. N. Vasilyev},
     title = {Mathematical modeling of distributed systems by neural networks},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {32--42},
     publisher = {mathdoc},
     volume = {19},
     number = {12},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2007_19_12_a3/}
}
TY  - JOUR
AU  - A. N. Vasilyev
TI  - Mathematical modeling of distributed systems by neural networks
JO  - Matematičeskoe modelirovanie
PY  - 2007
SP  - 32
EP  - 42
VL  - 19
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2007_19_12_a3/
LA  - ru
ID  - MM_2007_19_12_a3
ER  - 
%0 Journal Article
%A A. N. Vasilyev
%T Mathematical modeling of distributed systems by neural networks
%J Matematičeskoe modelirovanie
%D 2007
%P 32-42
%V 19
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2007_19_12_a3/
%G ru
%F MM_2007_19_12_a3
A. N. Vasilyev. Mathematical modeling of distributed systems by neural networks. Matematičeskoe modelirovanie, Tome 19 (2007) no. 12, pp. 32-42. http://geodesic.mathdoc.fr/item/MM_2007_19_12_a3/

[1] S. Khaikin, Neironnye seti: polnyi kurs, 2-e izdanie, Izd. dom “Vilyamc”, M., 2006

[2] A. I. Galushkin, Teoriya neironnykh setei, Kn. 1, IPRZhR, M., 2000

[3] Isaac E. Lagaris, Aristidis Likas, Dimitrios I. Fotiadis, “Artificial Neural Networks for Solving Ordinary and Partial Differential Equations”, IEEE Transactions on Neural Networks, 9:5 (1998), 987–1000 | DOI

[4] Ryusuke Masuoka, “Neural Networks Learning Differential Data”, IEICE Trans. Inf. and Syst., E83-D:8 (2000), 1291–1299

[5] A. I. Galushkin, Neiromatematika, Kn. 6, IPRZhR, M., 2002

[6] V. I. Gorbachenko, Neirokompyutery v reshenii kraevykh zadach teorii polya, Kn. 10, Radiotekhnika, M., 2003

[7] S. A. Terekhoff, N. N. Fedorova, Cascade Neural Networks in Variational Methods For Boundary Value Problems, Russian Federal Nuclear Center, VNIITF

[8] Edward J. Kansa, Motivation for using radial basis functions to solve PDEs, Lawrence Livermore National Laboratory and Embry-Riddle Aeronatical University, 1999; http://rbf-pde.uah.edu/kansaweb.ps

[9] Bengt Fornberg, Elisabeth Larsson, “A Numerical Study of some Radial Basis Function based Solution Methods for Elliptic PDEs”, Computers and Mathematics with Applications, 46 (2003), 891–902 | DOI | MR

[10] A. N. Vasilev, D. A. Tarkhov, “Primenenie neironnykh setei k neklassicheskim zadacham matematicheskoi fiziki”, Sbornik dokladov Mezhdunarodnoi konferentsii po myagkim vychisleniyam i izmereniyam SSM'2003, t. 1, SPb, SPbGETU “LETI”, 337–340

[11] A. N. Vasilev, D. A. Tarkhov, “Novye podkhody na osnove YuVR – setei k resheniyu kraevykh zadach dlya uravneniya Laplasa na ploskosti”, Neirokompyutery: razrabotka, primenenie, no. 7–8, Radiotekhnika, M., 2004, 119–126

[12] A. N. Vasilev, D. A. Tarkhov, “Neironnye seti kak novyi universalnyi podkhod k chislennomu resheniyu zadach matematicheskoi fiziki”, Neirokompyutery: razrabotka, primenenie, no. 7–8, Radiotekhnika, M., 2004, 111–118

[13] A. N. Vasilev, D. A. Tarkhov, “Neirosetevye podkhody k resheniyu kraevykh zadach v mnogomernykh sostavnykh oblastyakh”, Izvestiya TRTU, 2004, no. 9, 80–89

[14] A. N. Vasilev, D. A. Tarkhov, “Primenenie iskusstvennykh neironnykh setei k modelirovaniyu mnogokomponentnykh sistem so svobodnoi granitsei”, Izvestiya TRTU, 2004, no. 9, 89–100

[15] A. Vasilyev, D. Tarkhov, G. Guschin, “Neural Networks Method in Pressure Gauge Modeling”, Proceedings of the 10th IMEKO TC7 International Symposium on Advances of Measurement Science, V. 2 (2004, Saint-Petersburg, Russia), 275–279

[16] A. N. Vasilev, D. A. Tarkhov, “KVR-seti i nekotorye zadachi matematicheskoi fiziki”, Sbornik dokladov Mezhdunarodnoi konferentsii po myagkim vychisleniyam i izmereniyam-SSM'2004, T. 1, SPbGETU “LETI”, SPb, 309–312

[17] A. N. Vasilev, D. A. Tarkhov, “Neirosetevaya metodologiya postroeniya priblizhennykh reshenii differentsialnykh uravnenii po eksperimentalnym dannym”, Intellektualnye i mnogoprotsessornye sistemy-2005, Materialy mezhdunarodnoi konferentsii. T. 2, Taganrog-Donetsk-Minsk, 2005, 219–223

[18] A. N. Vasilyev, D. A. Tarkhov, “New neural network technique to the numerical solution of mathematical physics problems. I: Simple problems”, Optical Memory and Neural Networks, 14:1 (2005), 59–72 | MR

[19] A. N. Vasilyev, D. A. Tarkhov, “New neural network technique to the numerical solution of mathematical physics problems. I: Complicated and nonstandard problems”, Optical Memory and Neural Networks, 14:2 (2005), 97–122 | MR

[20] D. A. Tarkhov, Neironnye seti. Modeli i algoritmy, Kn. 18, Radiotekhnika, M., 2005

[21] A. N. Vasilev, D. A. Tarkhov, “Raschet teploobmena v sisteme “sosudy-tkani” na osnove neironnykh setei”, Sovremennye problemy neiroinformatiki, Ch. 2 kn. 23, Radiotekhnika, M., 2006

[22] A. N. Vasilev, “Novyi podkhod k postroeniyu priblizhennogo resheniya ellipticheskoi zadachi v sluchae mnogokomponentnoi oblasti na osnove iskusstvennykh neironnykh setei”, Materialy VI Mezhdunarodnoi konferentsii po neravnovesnym protsessam v soplakh i struyakh-NPNJ-2006, Vuzovskaya kniga, SPb.-M., 89–91

[23] A. N. Vasilev, D. A. Tarkhov, “Novyi podkhod k chislennomu resheniyu zadach matematicheskoi fiziki na osnove iskusstvennykh neironnykh setei”, Materialy VI Mezhdunarodnoi konferentsii po neravnovesnym protsessam v soplakh i struyakh-NPNJ-2006, Vuzovskaya kniga, SPb.-M., 91–92 | MR

[24] A. N. Vasilev, D. A. Tarkhov, “Evolyutsionnye algoritmy resheniya kraevykh zadach v oblastyakh, dopuskayuschikh dekompozitsiyu”, Matem. modelirovanie, 19:12 (2007), 52–62

[25] A. N. Vasilev, D. A. Tarkhov, “Postroenie priblizhennykh neirosetevykh modelei po raznorodnym dannym”, Matematicheskoe modelirovanie, 19:12 (2007), 43–51

[26] M. A. Aleksidze, Fundamentalnye funktsii v priblizhennykh resheniyakh granichnykh zadach, Nauka, M., 1991 | MR | Zbl

[27] L. Bers, F. Dzhon, M. Shekhter, Uravneniya s chastnymi proizvodnymi, Mir, M., 1966 | MR | Zbl

[28] A. N. Tikhonov, V. Ya. Arsenin, Metody resheniya nekorrektnykh zadach, Nauka, M., 1986 | MR | Zbl

[29] A. N. Vasilev, N. G. Kuznetsov, “O nekotorykh ekstremalnykh zadachakh, voznikayuschikh v akustike. Kraevye zadachi dlya neklassicheskikh uravnenii matematicheskoi fiziki”, Neklassicheskie uravneniya matematicheskoi fiziki, Sb. trudov vsesoyuznoi shkoly, Novosibirsk, 1989, 94–98 | MR