Abstract
In this paper, the inverse problem of technological thermophysics under the influence of disturbing factors is under study. In the problem of identifying the process of nonstationary heat conduction, it is required to concretize its mathematical model by qualitatively and quantitatively expressing an unknown characteristic based on the results of experimental studies. It is necessary to determine the uncontrolled time-varying heat flux density on the surface of the heated product from the noisy temperature measurement results at a certain point inside the object. The problem is formulated in an extreme setting as a problem of optimal control of an object with distributed parameters, in which the quadratic value of the temperature discrepancy between experimental and model data is used as an optimality criterion. The preliminary parametrization of the desired control on a compact set of polynomial functions implements the reduction to the parametric optimization problem. Physically substantiated solutions to inverse heat conduction problems are found as a result of their sequential parametric optimization using an algorithmically accurate method based on optimal control theory. The proposed solution combines the advantages of an accurate analytical method, which allows taking into account the physical essence of the process of interest and artificial intelligence methods, which provide great opportunities to find an quasioptimal solution under conditions of uncertainty in the mathematical description of the process. The analytical method of sequential parameterization provides a search for solutions on a compact set of smooth functions, as a result of which there is a reduction to the problem of parametric optimization. Measurement errors lead to processing large amounts of data, which necessitates the use of artificial neural networks for parametric optimization of the identified characteristics. The attained results confirm the possibility of obtaining adequate solutions to the inverse problems of thermal conductivity with the intensity of the measurement noise in the range of 0-15 %. In the investigated class of solutions, with a suitable setting of the ranges of belonging of the parameters, the error in approximating the temperature state can be up to 2-5 %, and the error in restoring the unknown characteristic can be up to 7-10 %.
Highlights
This work is devoted to the search for methods for solving inverse problems, formulated in relation to equations of mathematical physics of parabolic type and providing for the definition of an unknown function that sets the boundary conditions when the input data are not fully known
The main difficulty in solving inverse problems of mathematical physics (IPMP) is their belonging to the class of ill-posed problems, which necessitates the use of special numerical regularizing algorithms to obtain a stable solution [1], [2]
This paper presents an approach to the search for physically grounded solutions of the IPMP using the example of parabolic equations on compact sets of polynomial functions under the conditions of disturbed input data based on parametric optimization implemented using artificial neural networks (ANN)
Summary
This work is devoted to the search for methods for solving inverse problems, formulated in relation to equations of mathematical physics of parabolic type and providing for the definition of an unknown function that sets the boundary conditions when the input data are not fully known. In the theory of inverse ill-posed problems of mathematical physics, a large number of methods and computational algorithms have been developed that make it possible to obtain regular solutions based, as a rule, on the use of smoothing functionals or additional restrictions on the class of sought solutions [5]-[7]. These approaches require significant computational efforts, depending on factors that are difficult to formalize [1], [2] or the availability of a priori information about the solution, which is usually absent [1], [4]. Difficulties associated with the need to process a large amount of input information affect the training of an artificial neural network, and practically do not affect its performance when it is subsequently used to solve a specific problem of identifying the boundary conditions of the process
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