Construction of universal neural network solutions for families of boundary value problems of the piezoelectric equation

About the authors:

Roman Yu. Ponomarev, Manager, Tyumen Petroleum Research Center, Tyumen, Russia; Postgraduate Student, Department of Modeling of Physical Processes and Systems, School of Natural Sciences, University of Tyumen, Tyumen, Russia; ryponomarev@tnnc.rosneft.ru



Mikhail I. Ivlev, Chief Specialist, RN-Geology Research and Development, Tyumen, Russia; miivlev@ rn-gir.rosneft.ru

Ruslan R. Migmanov, Chief Specialist, RN-Geology Research and Development, Tyumen, Russia; rrmigmanov @rn-gir.rosneft.ru

Vladimir E. Vershinin, Chief Specialist, Tyumen Petroleum Research Center, Tyumen, Russia; Associate Professor, Department of Physical Processes and Systems Modeling, School of Natural Sciences, University of Tyumen, Tyumen, Russia; ve_vershinin2@tnnc.rosneft.ru

Abstract:

The article is devoted to the development of methods for obtaining neural network analogues of analytical solutions to families of boundary value problems for partial differential equations. Approximation of solutions to boundary value problems of differential equations is an urgent task of mathematical modeling and an integral element of optimization algorithms. There are known methods for obtaining analytical approximations of solutions to boundary value problems by training physical informed neural networks. The fulfillment of the equation and the initial boundary conditions during the training of such a neural network is ensured by adding the corresponding terms of the discrepancy to the target learning functional. This approach leads to the fact that when training a neural network, only the pre-set initial boundary conditions are taken into account, and when they change, new training is required. This significantly increases the calculation time and limits the applicability of physically informed neural networks in solving optimization problems.

The paper proposes a method for taking into account a parameterized set of initial boundary conditions when training a physically informed neural network. Using the example of boundary value problems with arbitrary initial and boundary conditions for the diffusivity equation with variable coefficients, a physical informed neural network operator is constructed that allows obtaining families of approximation solutions and an estimate of their accuracy is given. The proposed method is based on changing the structure of the neural network and using variable weighting coefficients instead of constant ones, which makes such a neural network a kind of Kolmogorov–Arnold networks. It is proposed to use nested perceptron neural networks to find the weight functions. The proposed approach makes it possible to solve optimization problems using trained physically informed neural networks without resorting to numerical grid solutions.

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