Page 205 - Neural Network Modeling and Identification of Dynamical Systems

196 5. SEMIEMPIRICAL NEURAL NETWORK MODELS OF CONTROLLED DYNAMICAL SYSTEMS

ing examples are located in a small area of the research. Studies in computational intelligence, vol. 799.
input space has a similar effect as the assign- Berlin: Springer Nature; 2019. p. 196–201.
ment of the weight ¯n to some mean example [7] Tarkhov DA, Vasilyev AN. New neural network tech-
nique to the numerical solution of mathematical physics
from this area. Thus, the nonuniform distribu- problems. I: Simple problems. Opt Memory Neural
tion of training examples in U × X might lead Netw (Inf Opt) 2005;14(1):59–72.
to higher model accuracy in some areas of the [8] Tarkhov DA, Vasilyev AN. New neural network tech-
input space at the expense of the lower accu- nique to the numerical solution of mathematical physics
problems. II: Complicated and nonstandard problems.
racy in other areas. In order to avoid this effect,
Opt Memory Neural Netw (Inf Opt) 2005;14(2):97–122.
we need to perform appropriate weighting for [9] Kainov NU, Tarkhov DA, Shemyakina TA. Application
individual training examples. In particular, the of neural network modeling to identification and pre-
weight of a training example might be taken diction problems in ecology data analysis for metal-
inversely proportional to the number of train- lurgy and welding industry. Nonlinear Phenom Com-
plex Syst 2014;17(1):57–63.
ing examples in its neighborhood of fixed ra- [10] Vasilyev AN, Tarkhov DA. Mathematical models
dius. of complex systems on the basis of artificial neu-
An efficient software implementation of the ral networks. Nonlinear Phenom Complex Syst
algorithms described in this section should take 2014;17(3):327–35.
advantage of a special data structure for the [11] Budkina EM, Kuznetsov EB, Lazovskaya TV,
Tarkhov DA, Shemyakina TA, Vasilyev AN. Neu-
storage of training set points in order to pro- ral network approach to intricate problems solving for
vide fast operations of the nearest neighbor ordinary differential equations. Opt Memory Neural
search as well as the search for all neighbors Netw (Inf Opt) 2017;26(2):96–109.
in a fixed radius. One reasonable candidate [12] Lazovskaya TN, Tarkhov DA, Vasilyev AN. Paramet-
ric neural network modeling in engineering. Recent
is a k-d tree structure that allows for approx-
Patents Eng 2017;11(1):10–5.
imate nearest neighbor searches, such as the [13] Lazovskaya TN, Tarkhov DA, Vasilyev AN. Multi-
one implemented in the FLANN library [56, layer solution of heat equation. Stud Comput Intell
57]. 2018;736:17–22.
[14] Vasilyev AN, Tarkhov DA, Tereshin VA, Berminova MS,
Galyautdinova AR. Semi-empirical neural network
model of real thread sagging. Stud Comput Intell
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