NEURAL NETWORK METHOD OF RESTORING AN INITIAL PROfiLE OF THE SHOCK WAVE

• Main Authors: Tatiana A. Shemyakina, Dmitriy A. Tarkhov, Alexandra R. Beliaeva, Ildar U. Zulkarnay
• Format: Article
• Language: Russian
• Published: The Fund for Promotion of Internet media, IT education, human development «League Internet Media» 2018-03-01
• Series: Современные информационные технологии и IT-образование
• Subjects: Inverse problem; neural networks; shock wave
• Online Access: http://sitito.cs.msu.ru/index.php/SITITO/article/view/355

In this paper, we apply neural network modeling to solve the inverse problem of mathematical physics with a system of nonlinear partial differential equations of hyperbolic type. In the problem, the initial conditions are unknown and are reconstructed from measurements made at a later point in time. For this we use the methodology developed by us to construct approximate mathematical models with respect to differential equations and additional data. It is known that inverse problems are difficult to apply classical numerical methods for solving boundary value problems for partial differential equations and require the use of various artificial methods. Our approach allows us to solve both direct and inverse problems in almost the same way. We reconstruct the initial profile of the pressure distribution in the tube in which the shock wave propagates, as measured by the sensor at the end of the tube. In this neural network model we use a perceptron with one hidden layer with an activation function in the form of a hyperbolic tangent. It is known that such a neural network is a universal approximator, i.e. allows us to arbitrarily accurately approximate a function from a sufficiently wide class (in particular, the desired solution of the problem belongs to this class). We also tried other architectures of neural networks, in particular, a network with radial basis functions (RBF), but for this task they were less suitable. Previously, we applied this approach to problems with a known analytical solution in order to verify the results of the application of the method. Our method proved to be sufficiently accurate and robust to errors in the original data. A feature of this work is the application of the method to the problem with real measurements. The obtained results allow us to recommend the proposed method for solving other similar problems.