Neural Network Technique in Boundary Value Problems for Ordinary Differential Equations

Budkina, E. M.; Kuznetsov, E. B.; Lazovskaya, Tatiana; Leonov, Sergey S.; Tarkhov, Dmitriy; Vasilyev, Alexander

doi:10.1007/978-3-319-40663-3_32

Cited by 12 publications

(10 citation statements)

References 6 publications

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“…These solutions should allow for the possibility of refinement according to monitoring data of the object. The complex of our methods for constructing approximate neural network solutions is described and tested on a variety of problems for ODE and PDE, [1][2][3][4][5][6][7]. In particular, methods of adjusting models to new data are presented.…”

Section: Introductionmentioning

confidence: 99%

Comparison of two neural network approaches to modeling processes in a chemical reactor

et al. 2019

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In this paper, we conduct the comparative analysis of two neural network approaches to the problem of constructing approximate neural network solutions of non-linear differential equations. The first approach is connected with building a neural network with one hidden layer by minimization of an error functional with regeneration of test points. The second approach is based on a new continuous analog of the shooting method. In the first step of the second method, we apply our modification of the corrected Euler method, and in the second and subsequent steps, we apply our modification of the Störmer method. We have tested our methods on a boundary value problem for an ODE which describes the processes in the chemical reactor. These methods allowed us to obtain simple formulas for the approximate solution of the problem, but the problem is special because it is highly non-linear and also has ambiguous solutions and vanishing solutions if we change the parameter value.

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Section: Introductionmentioning

confidence: 99%

Comparison of two neural network approaches to modeling processes in a chemical reactor

et al. 2019

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“…We will explain the essence of the neural network approach that we have developed by using the example of the simplest boundary‐value problem

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…”

Section: Introductionmentioning

confidence: 99%

“…Nonstationary problems (specifically initial boundary value problems) can be considered in the context of this approach by changing the space dimension, ie, by including time in the set of variables. However, there are other ANN‐approaches; the application of dynamic neural networks, in particular, see in related works …”

Section: Introductionmentioning

confidence: 99%

“…This stage is based on the information about the models of the phenomena being studied (these models can be refined, whereas the solution is being searched for; while the object is being constructed and when it is in operation). The choice of the functional baseline (baselines) . This stage can be performed both by an expert in the subject area on the basis of information about the nature of the phenomena being simulated and automatically by using evolutionary algorithms (see other works). We suggest using the function bases typical for neural networks .…”

Section: Introductionmentioning

confidence: 99%

“…This stage can be performed both by an expert in the subject area on the basis of information about the nature of the phenomena being simulated and automatically by using evolutionary algorithms (see other works). We suggest using the function bases typical for neural networks . It appeared that neural network functions of the same types could be applied to solve various problems; they are tolerant to the errors in the original data, and they contribute to efficient software and hardware implementation. The choice and implementation of the methods for the selection of parameters and the model structure .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Unified approach to constructing the neural network models of real objects. Part 1

Antonov

Tarkhov

Vasilyev

2018

Math Methods in App Sciences

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This article is a two-part composition. It describes the basics of the methodology developed by the authors to construct approximate neural network solutions of ordinary and partial differential equations. The methodology is illustrated with the sample tasks, ie, for compound domains (with a different type of equations in the subdomains) in the first part, for a variable boundary (caused by phase transition), for a selected boundary, with some heterogeneous information, which includes differential equations, initial and boundary conditions, and experimental and other data in the second part. We show that these different tasks are solved uniformly with the application of general principles. KEYWORDSartificial neural network, differential equation, error functional, heterogeneous data, initial-boundary value problem, network learning 9244

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From Differential Equations to Multilayer Neural Network Models

Kaverzneva

Malykhina

Tarkhov

2019

Advances in Neural Networks – ISNN 2019

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Neural Network Technique in Boundary Value Problems for Ordinary Differential Equations

Cited by 12 publications

References 6 publications

Comparison of two neural network approaches to modeling processes in a chemical reactor

Comparison of two neural network approaches to modeling processes in a chemical reactor

Unified approach to constructing the neural network models of real objects. Part 1

From Differential Equations to Multilayer Neural Network Models

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