Numerical solving of hydrodynamic problems using machine learning methods

Abstract

The use of the neural network approach in the process of finding a solution of a hydrodynamic problem by numerical methods is proposed. The two areas of possible application of the technology of neural networks are considered – the choice of initial approximation to the solution and the search for the next approximation. To select the initial approximation, it is proposed to solve the combined task of classification and regression based on the existing base of distributions samples and on the existing base of patterns of space transformations. An architecture of the combined neural network which solves this problem is proposed. For finding the transformation of the problem space, the use of radial-basis neural network is proposed. The mathematical apparatus for tuning the network with arbitrary number of neurons of the output layer is proposed. It is proposed to take the number of hidden layer neurons less than the number of educational transformation samples and find an approximate solution. The task of finding the optimal weight values in this case can be considered as a task of minimization of the target function, which describes the network outputt error. It is proposed to construct a neural network for finding the next approximation of a solution of a hydrodynamic problem based on a generalization of the principle previously proposed for solving solutions of one-dimensional differential equations. Finding the next approximation in the case of solving a task on a multiprocessor system is presented as a game with multiple players, each of which must find a compromise between local and global search purposes. It is proposed to replace one common neural network with a set of neural networks that interact with each other. The proposed approaches can reduce the amount of computation needed to find a solution.