Application of neural network modeling methods in solving initial boundary value problems for partial differential equations

About the authors:

Vladimir E. Vershinin, Associate Professor, Department of Physical Processes and Systems Modeling, University of Tyumen; eLibrary AuthorID, Scopus AuthorID, v.e.vershinin@utmn.ru

Roman Yu. Ponomarev, Chief Specialist, Tyumen Petroleum Research Center, Tyumen, Russia, ryponomarev@tnnc.rosneft.ru

Abstract:

Machine learning allows you to solve a variety of data analysis problems, but its use for solving differential equations has appeared relatively recently. The approximation of the solution of the boundary value problem for differential equations (ordinary and partial derivatives) is constructed using neural network functions. The selection of weighting coefficients is carried out during the training of the neural network. The criteria for the quality of training in this case are inconsistencies in the equation and boundary-initial conditions. This approach makes it possible, instead of grid solutions, to find solutions defined on the entire feasible region of the boundary value problem. Specific examples show the features of the application of physics-informed neural networks to the solution of boundary value problems for differential equations of various types. Physics-informed neural networks training methods can be used in the tasks of retraining intelligent control systems on incomplete sets of input data.

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